Document [original]

Spaces of continuous and holomorphic

functions with growth conditions

Simone Agethen

Mathematical Institute

University of Paderborn

November 2004

Dedicated to my father...

...who believed in me, whatever I did.

It was a great pleasure to write my thesis under the direction of Klaus Bier-

stedt and Jos´e Bonet. I would like to thank them for many interesting

discussions, ideas and all the support.

I would like to thank Wolfgang Lusky for many helpful discussions and his

ideas, especially in connection with the second part of this work.

A sincere thank you also to the following persons for their help in many

different ways: B. Duddeck, B. Ernst, W. Kaballo, J. Wengenroth, E. Wolf,

D. Vogt.

Last, but not least, I would like to thank my family for all their non-

mathematical help, their patience and just for being there, whenever I needed

them.

Contents

1 Weighted (PLB)-spaces of continuous functions 5

1.1 Introduction to part 1 . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Notation and definitions . . . . . . . . . . . . . . . . . . . . . 8

1.3 Consequences of known results . . . . . . . . . . . . . . . . . . 9

1.4 The (LF)-case . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4.1 General results for (LF )-spaces . . . . . . . . . . . . . 10

1.4.2 Conditions (Q) and (wQ) . . . . . . . . . . . . . . . . 12

1.4.3 The inductive limits VC(X) and V0C(X) . . . . . . . 14

1.5 Known results of projective spectra of (DF)-spaces . . . . . . 18

1.6 New results on the weighted (PLB)-spaces A0C(X) and AC(X) 25

1.6.1 Structure of A0C(X) . . . . . . . . . . . . . . . . . . 25

1.6.2 Structure of AC(X) . . . . . . . . . . . . . . . . . . . 31

1.7 Inductive description . . . . . . . . . . . . . . . . . . . . . . . 33

1.7.1 Inductive description for Fr´echet spaces . . . . . . . . . 33

1.7.2 Inductive description in the (PLB)-case . . . . . . . . . 34

1.8 Comparison of the (P LB)- and the (LF )-space . . . . . . . . 39

1.9 An example in the case of sequence spaces . . . . . . . . . . . 43

2 Weighted spaces of holomorphic functions on the half-plane 47

2.1 Introduction to part 2 . . . . . . . . . . . . . . . . . . . . . . 47

2.2 Notation and known results . . . . . . . . . . . . . . . . . . . 48

2.3 Commuting b.a.p. and the main result . . . . . . . . . . . . . 51

2.4 Preparations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.5 Proof of theorem 2.13 . . . . . . . . . . . . . . . . . . . . . . . 61

2.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

References 66

1 Weighted (PLB)-spaces of continuous func-

tions

1.1 Introduction to part 1

In the first chapter of this work we investigate the weighted (P LB)-spaces

AC(X) and A0C(X) of continuous functions, i.e. for a double sequence

A:= ((an,k)k∈N)n∈Nof strictly positive continuous functions (weights) with

an,k+1(x)≤an,k(x)≤an+1,k(x)∀n, k ∈N, x ∈X, we form the projective

limit (with respect to n) of the inductive limits (with respect to k) of the

weighted spaces of continuous functions Can,k(X) and C(an,k)0(X), respec-

tively.

Weighted spaces of continuous functions were introduced by Nachbin ([33],

[34], [35]). Inductive limits of weighted spaces of continuous and holomorphic

functions were studied by Bierstedt, Meise [12] in 1976. In 1982 Bierstedt,

Meise, Summers [14] investigated the projective description of weighted in-

ductive limits. They showed that the weighted inductive limit V0C(X) is

always a topological subspace of its projective hull CV 0(X) and that in

the (LB)-case V0C(X) is complete if and only if V0C(X) = CV 0(X) holds

algebraically (and topologically) if and only if the sequence Vis regularly

decreasing. For O-growth conditions Bierstedt, Bonet [6] and Bastin [2] gave

a similar result in 1989: For a locally compact and σ-compact space Xthe

(LB)-space VC(X) equals CV (X) topologically (and algebraically) if and

only if the sequence Vsatisfies condition (D) (compare section 1.4.3).

The more complicated case of weighted (LF )-spaces of continuous functions

was investigated by Bierstedt, Bonet [8] in 1994. For weighted (LF)-spaces

of continuous functions they used the conditions (Q) and (wQ) of Vogt to

obtain results for the projective description.

In the following chapter we investigate the weighted (PLB)-spaces AC(X)

and A0C(X) of continuous functions for the first time. We will analyse

their topological structures and for o-growth conditions we can characterise

when the (PLB)-space A0C(X) and the (LF )-space V0C(X) are equal alge-

braically and topologically.

In section 1.2 we give the necessary notations and definitions for the first

chapter. In section 1.3 we collect some properties of the spaces AC(X) and

A0C(X) which follow from the general theory of Banach spaces and of their

inductive and projective limits. After that we recall in section 1.4 several

results for general (LF )-spaces: In 1.4.1, we give the definitions of acyclic

and weakly acyclic (LF )-spaces in the sense of Palamodov [36], recall the

characterisation of Retakh [38] and present the characterisation of Vogt [44]

with the conditions (Q) and (wQ) as well as some results of Wengenroth

[46]. Next, in 1.4.2 we introduce the conditions (Q) and (wQ) of Vogt in

the way Bierstedt, Bonet [8] used them to investigate weighted (LF )-spaces

of continuous functions. They reformulated them in terms of the weights,

introduced a condition (wQ∗) which is equivalent to (wQ), and constructed

many examples of which we present some here, too. At the end of this section

in 1.4.3 we collect the main results for the weighted (LB)- and (LF )-spaces

VC(X) and V0C(X) and their projective description which were proved by

Bastin [2], Bierstedt, Bonet [8] and Bierstedt, Meise, Summers [14]. Then

we present the general theory of projective spectra of (DF)-spaces in section

1.5. These results go back to Palamodov, Retakh, Vogt and Wengenroth

([36], [38], [42], [43], [45]).

After all this we are finally able to investigate the structure of the weighted

(PLB)-spaces AC(X) and A0C(X). In 1.6.1 we show for the weighted

(PLB)-spaces A0C(X) that condition (wQ) is equivalent to Proj1A0= 0

and that it is also equivalent to A0C(X) ultrabornological (theorem 1.48).

Furthermore we prove that the projective spectrum A0is of strong P-type if

and only if condition (Q) is satisfied (theorem 1.49). In the case of O-growth

conditions we show in section 1.6.2 that Proj1A= 0 if and only if condition

(Q) is satisfied (theorem 1.52) if and only if the projective spectrum Ais of

strong P-type (remark 1.53).

Next follow the inductive description for weighted Fr´echet spaces in section

1.7.1 and the inductive description in the case of weighted (P LB)-spaces

in section 1.7.2. In both cases we prove that for a locally compact and σ-

compact space Xthe spaces CA(X) and CA(X) resp. AC(X) are equal

algebraically (theorem 1.55 and 1.56). In the case of O-growth conditions it

was not possible to give a similar characterisation as in theorem 1.48. But

the inductive description allows us to conclude that from AC(X) barrelled

it follows that condtion (wQ) is satisfied (see corollary 1.60 and remark 1.61).

In section 1.8 we compare the weighted (P LB)-spaces of continuous func-

tions with the weighted (LF)-spaces of continuous functions. First we give

an example which shows that these spaces are not equal in general. Then

we introduce condition (B) of Vogt and prove that AC(X) = VC(X) holds

algebraically if and only if condition (B) is satisfied (theorem 1.65). To prove

the same in the case of o-growth condtions we need the additional condition

that (An)0C(X) is complete for each n∈N(theorem 1.66). If all (An)0C(X)

are complete we can even prove that A0C(X) = V0C(X) holds algebraically

and topologically if and only if the conditions (B) and (wQ) are satisfied

(corollary 1.67).

We finish this chapter in section 1.9 with an example in the case of sequence

spaces which illustrates the results given above.

1.2 Notation and definitions

Weighted (PLB)-spaces of continuous functions are defined by a double se-

quence of weights as the projective limit of the inductive limit of the single

weighted Banach spaces of continuous functions. In this work these spaces

are investigated for the first time. Taking the limits the other way round

one gets the (LF)-spaces of continuous functions which were investigated by

Bierstedt, Bonet [8]. From now on let Xbe a locally compact space and

A:= ((an,k)k∈N)n∈Na double sequence of strictly positive continuous func-

tions on X, called weights, which is decreasing in kand increasing in n, i.e.

for each n, k ∈Nand x∈X

an,k+1(x)≤an,k(x)≤an+1,k(x)

holds. Define the weighted spaces of continuous functions

Can,k(X) := {f∈C(X) ; ||f||n,k := sup

x∈X

an,k(x)|f(x)|<∞},

C(an,k)0(X) := {f∈C(X) ; an,k|f|vanishes at infinity on X}

with continuous inclusions

Can,k(X)→Can,k+1(X) and C(an,k)0(X)→C(an,k+1)0(X)

for fixed n∈N.

C(an,k)0(X) is a closed subspace of Can,k(X), and both spaces are complete,

hence Banach spaces, where C(an,k)0(X) carries the induced norm. The unit

balls are

Bn,k := {f∈Can,k(X); an,k|f| ≤ 1},

(Bn,k)0:= {f∈C(an,k)0(X); an,k|f| ≤ 1}.

We form the locally convex inductive limits

AnC(X) := indkCan,k(X),

(An)0C(X) := indkC(an,k)0(X)

with An+1C(X)⊂ AnC(X) and (An+1)0C(X)⊂(An)0C(X) and the pro-

jective limits

AC(X) := projnAnC(X) = projnindkCan,k(X),

A0C(X) := projn(An)0C(X) = projnindkC(an,k)0(X).

1.3 Consequences of known results

In this section we collect known results for the weighted (PLB)-spaces AC(X)

and A0C(X) which follow from the general theory of Banach resp. Fr´echet

spaces and their countable inductive and projective limits. Every Fr´echet

space Fas well as every Banach space Eis webbed [32], which means that

there exists a family Cn1,...,nk, n1, ..., nk∈N, k ∈N, of absolutely convex

subsets of Fwith the following properties:

i) ∪∞

n=1Cn=F.

ii) ∪∞

n=1Cn1,...,nk,n =Cn1,...,nkfor all n1, ..., nk∈Nand all k∈N.

iii) For each sequence (nk)k∈Nin Nthere exists a sequence (λk)k∈Nin ]0,∞[,

so that for every sequence (xk)k∈Nin Fwith xk∈Cn1,...,nkfor all k∈N

the series P∞

k=1 λkxkconverges in F.

Remark 1.1. The spaces C(an,k)(X) and C(an,k)0(X) are webbed for each

k, n ∈N.

To show that AC(X) and A0C(X) are webbed we need some results of de

Wilde [47]:

Theorem 1.2. i) A countable inductive limit of webbed spaces is webbed.

ii) A countable projective limit of webbed spaces is webbed.

As a consequence of theorem 1.2 we get:

Corollary 1.3. The spaces AnC(X)and (An)0C(X)are webbed for each

n∈N. The spaces AC(X)and A0C(X)are webbed.

Fr´echet and Banach spaces are ultrabornological (and hence barrelled) [32].

Remark 1.4. The spaces C(an,k)0(X) and C(an,k)(X) are ultrabornological

(and hence barrelled) for each n, k ∈N.

Theorem 1.5. (Meise, Vogt [32]) Let the locally convex space Ecarry the

inductive topology of the system (ji:Ei→E)i∈I. If all the spaces Eiare

barrelled or ultrabornological, then Ehas the corresponding property, too.

Corollary 1.6. The spaces (An)C(X)and (An)0C(X)are ultrabornological

(and hence barrelled) for each n∈N.

In general the countable projective limit of barrelled resp. ultrabornological

spaces need not be barrelled resp. ultrabornological. Conditions under which

AC(X) and A0C(X) are barrelled resp. ultrabornological will be discussed

later.

1.4 The (LF )-case

Palamodov [36] investigated acyclic and weakly acyclic (LF)-spaces with ho-

mological tools. Retakh [38] later called them (LF)-spaces of type (M) and

(M0). In 1992 Vogt investigated these spaces with more functional analytic

tools and reformulated the conditions of Retakh. Vogt introduced the con-

ditions (Q) and (wQ) as necessary conditions for acyclic and weakly acyclic

(LF)-spaces. In this section we collect some of these results to compare them

with the (PLB)-spaces and with the results we will obtain here.

1.4.1 General results for (LF )-spaces

Let Ebe an (LF )-space. This means here that there is an increasing sequence

(Ek)k∈Nof subspaces of Ewith continuous imbeddings, each Ekis equipped

with a Fr´echet space topology, and E:= indkEk. For each k∈Nthere is a

fundamental system (|| ||k,n)nof seminorms in Ekand we assume that

|| ||k+1,n ≤ || ||k,n ≤ || ||k,n+1

holds for each k, n ∈N. Identifying ⊕kEkwith the set {x= (xk)k∈

Qk∈NEk;xk= 0 up to finitely many k}we define the map q:⊕kEk→E

by q(x) = Pk∈Nxk.σ:⊕k∈NEk→ ⊕k∈NEk, defined by σ(x) = (xk−

xk−1)k, x−1= 0, is an isomorphism onto the kernel of q. Hence we have the

canonical exact sequence

0→ ⊕k∈NEk

−→ ⊕k∈NEk

−→E→0.

The inverse σ−1: ker q→ ⊕k∈NEkis given by

σ−1(x) = (

j=1

xj)k.

qis continuous and open, σis continuous, but not necessarily open onto its

range. This means that σ−1need not be continuous.

Definition 1.7. (Palamodov [36]) An inductive spectrum is called acyclic

if σ−1is continuous; it is called weakly acyclic if σ−1is weakly continuous,

i.e. continuous with respect to the weak topologies. An (LF)-space Eis

called (weakly) acyclic if it admits an acyclic resp. a weakly acyclic defining

spectrum.

Retakh [38] investigated this behaviour of an inductive spectrum and proved:

Theorem 1.8. Eis (weakly) acyclic if and only if the following condition is

fulfilled: There exists a sequence Uµof absolutely convex neighbourhoods of

zero in E, µ = 0,1,2, ..., such that

i) Uµ⊂Uµ+1 for all µ.

ii) For every µthere exists K≥µsuch that for all K≥kthe (weak)

topology of EKcoincides on Uµwith the (weak) topology of Ek.

In the notation of Retakh [38] the condition in the acyclic case is called (M),

in the weakly acyclic case (M0). Vogt showed that the following conditions

(Q) and (wQ) are necessary for acyclicity resp. weak acyclicity.

Proposition 1.9. i) If Eis acyclic, then condition (Q)holds:

∀n∃m≥n, k ∀µ≥m, l, ε > 0∃L, S > 0∀x∈En:

||x||m,l ≤ε||x||n,k +S||x||µ,L.

ii) If Eis weakly acyclic, then condition (wQ)holds:

∀n∈N∃m≥n, k ∈N∀µ≥m, l ∈N∃L∈N, S > 0∀x∈En:

||x||m,l ≤S(||x||n,k +||x||µ,L).

These conditions can be evaluated in concrete cases and are more suitable

for applications than the characterisations of Retakh.

Definition 1.10. An inductive limit E= indnEnis called regular if every

bounded subset of Eis contained and bounded in some step En.

It is well-known that every complete (LF)-space is regular, but whether the

converse holds is an open problem (raised by Grothendieck), even for (LB)-

spaces.

Definition 1.11. Let (E, t) = indn(En, tn) be an (LF)-space. The inductive

limit (E, t) is called sequentially retractive if every convergent sequence in

(E, t) is contained and convergent in some step (En, tn). It is called boundedly

retractive if for any bounded subset Bof (E, t) there is n∈Nsuch that Bis

contained and bounded in (En, tn) and that the topologies tand tncoincide

on B. The inductive limit (E, t) is called (sequentially) compactly regular if

every (sequentially) compact subset of the inductive limit is (sequentially)

compact in some step.

Palamodov [36] showed that every acyclic (LF)-space is complete, regular

and sequentially retractive. In 1996 Wengenroth [46] showed that for a gen-

eral (LF)-space the conditions (M) and (Q) are equivalent to the properties

of being sequentially retractive, boundedly retractive, compactly regular and

sequentially compactly regular.

1.4.2 Conditions (Q)and (wQ)

Vogt [44] introduced the conditions (Q) and (wQ) for general acyclic and

weakly acyclic (LF )-spaces, but we do not need these general conditions in

the sequel. In the case of weighted (PLB)-spaces one can reformulate these

conditions in terms of the weights, as follows:

Definition 1.12. i) A sequence A= ((an,k)k∈N)n∈Nsatisfies condition

(Q) if and only if

∀n∃m≥n, k ∀µ≥m, l, ε > 0∃L, S > 0∀x∈X:

am,l(x)≤max(ε1

an,k(x), S 1

aµ,L(x)).

ii) A sequence A= ((an,k)k∈N)n∈Nsatisfies condition (wQ) if and only if

∀n∈N∃m≥n, k ∈N∀µ≥m, l ∈N∃L∈N, S > 0∀x∈X:

am,l(x)≤Smax( 1

an,k(x),1

aµ,L(x)).

Note that condition (wQ) is always satisfied in the (LB)-case.

Bierstedt and Bonet introduced a condition similar to (wQ) which they called

condition (wQ∗).

Definition 1.13. (Bierstedt, Bonet [8]) A sequence A= ((an,k)k∈N)n∈Nsat-

isfies condition (wQ∗) if

∃(k(ν))ν∈N∀n∃m≥n∀µ≥m, l ∃L, S > 0∀x∈X:

am,l(x)≤Smax( min

1≤ν≤n

aν,k(ν)(x),1

aµ,L(x)).

Lemma 1.14. (Bierstedt, Bonet [8]) Condition (wQ)is equivalent to con-

dition (wQ∗).

At the end of this section we will present some examples of sequences of

weights which satisfy the conditions introduced above.

Example 1.15. (Bierstedt, Bonet [8]) Let

v:X→R, 0 < v(x)≤1∀x∈X,

w:X→R, 0 ≤w(x)∀x∈X,

be continuous functions, r, ρ > 0 or +∞, and let (rn)n∈N,(ρk)k∈Nbe strictly

increasing sequences of positive numbers with rn→rand ρk→ρ. For each

n, k ∈Nwe put

vn,k(x) := v(x)rnw(x)ρk∀x∈X

and V:= ((vn,k)k∈N)n∈N. If ρ=∞, then the sequence V= ((vn,k)k∈N)n∈N

satisfies condition (Q) and therefore (wQ).

Next we give an example of a sequence V= ((vn,k)k∈N)n∈Nwhich satisfies

condition (wQ), but not (Q). First we have to define regularly decreasing

sequences in the sense of Bierstedt, Meise, Summers [14]:

Definition 1.16. Let V= (vn)n∈Nbe a decreasing sequence of weights on

X.Vis called regularly decreasing if, for given n∈N, there exists m≥n,

such that for every ε > 0 and every k≥m, it is possible to find δ(k, ε)>0

with

vk(x)≥δ(k, ε)vn(x) whenever vm(x)≥εvn(x).

In other words, Vis regularly decreasing if and only if for given n∈N, there

exists m≥nsuch that, on each subset of Xon which the quotient vm

vnis

bounded away from zero, also all quotients vk

vn, k ≥m, are bounded away

from zero.

Example 1.17. (Bierstedt, Meise, Summers [15]) Let X:= N×N. The

sequence

vn(i, j) = (1

jii≤n

jni≥n+ 1 ,(i, j)∈N×N

is regularly decreasing, and the sequence

vn(i, j) = 1

jni≤n−1

ini≥n,(i, j)∈N×N

is not regularly decreasing.

Now to the example of a sequence V= ((vn,k)k∈N)n∈Nwhich satisfies condition

(wQ), but not (Q).

Example 1.18. (Bierstedt, Bonet [8]) Let W= (wn)n∈Nbe a decreasing

sequence of weights on a locally compact space Xwhich is not regularly

decreasing. For

vn,k = 2kwn, n, k ∈N,

V= ((vn,k)k∈N)n∈Nsatisfies condition (wQ), but not (Q).

Remark 1.19. (Bierstedt, Bonet [8])

(Q)⇔(wQ) plus the 00countably regularly decreasing00 condition (cRD) :

∀n∃m, k ∀µ, l, ε > 0∃L, δ > 0 :

vm,l(x)≥εvn,k(x)⇒vµ,L(x)≥δvn,k(x).

Hence in the (LB)-case condition (Q) is equivalent to the regularly decreasing

condition.

1.4.3 The inductive limits VC(X)and V0C(X)

In this section we will give a survey on inductive limits of weighted Banach

and Fr´echet spaces and their projective description. In the case of (LF)-

spaces V0C(X) Bierstedt, Meise, Summers [15] showed that the topology of

the weighted inductive limit V0C(X) can always be described by an associ-

ated system Vof weights on X.

In the beginning of this section we will restrict our attention to the (LB)-

case; the (LF)-case will be treated later on.

For a decreasing sequence V= (vn)n∈Nof strictly positive continuous func-

tions (weights) define

Cvn(X) := {f∈C(X); ||f||n= sup

x∈X

vn(x)|f(x)|<∞},

C(vn)0(X) := {f∈C(X); vnfvanishes at ∞on X},

and the weighted inductive limits of spaces of continuous functions

VC(X) := indnCvn(X) and V0C(X) := indnC(vn)0(X).

The associated system Vof weights was introduced by

V:= {v∈C(X); ∀n∃αn>0, v ≤inf

nαnvnon X}.

The corresponding weighted spaces for V(the projective hulls) are

CV (X) := {f∈C(X); ∀v∈V: sup

x∈X

v(x)|f(x)|<∞}

and

CV 0(X) := {f∈C(X); ∀v∈V:v|f|vanishes at ∞on X}.

CV (X) and CV 0(X) are complete, and C(V)0(X) is a closed subspace of

CV (X). For the case of o-growth conditions Bierstedt, Meise, Summers [14]

showed in 1982:

Theorem 1.20. In the (LB)-case of V0C(X), the following conditions are

equivalent:

i) V= (vn)n∈Nis regularly decrasing, i.e. for given n∈N, there exists

m≥n, such that for every ε > 0and every k≥m, it is possible to

find δ(k, ε)>0with

vk(x)≥δ(k, ε)vn(x) whenever vm(x)≥εvn(x),

ii) V0C(X)is complete,

iii) V0C(X) = CV 0(X)holds algebraically (and then also topologically).

Before we can formulate a result for O-growth conditions we have to introduce

condition (D), which was first used by Bierstedt, Meise [13] as a sufficient

condition for distinguishedness of echelon spaces. This property generalizes

the quasinormable and the reflexive case of echelon spaces. It was inspired

by a condition of Grothendieck [23], see also [15].

Definition 1.21. The decreasing sequence V= (vn)n∈Nsatisfies condition

(D) if there exists an increasing sequence J= (Xm)m∈Nof subsets Xmof X

such that

(N, J) for each m∈Nthere is nm≥mwith infx∈Xm

vk(x)

vnm(x)>0, k =nm+

1, nm+ 2, ..., while

(M, J) for each n∈Nand each subset Yof Xwith Y∩(X\Xm)6=∅for

all m∈Nthere is n0=n0(n, Y )> n such that infy∈Yvn0(y)

vn(y)= 0.

In the following result we need the assumption that the space Xis not only

locally compact, but also σ-compact. A characterisation of σ-compactness of

a locally compact space Xwas given by Bastin [1]. The condition (M, K) was

introduced by Bierstedt, Meise [13]. Condition (M1,K) and the continuous

domination property were defined by Bastin [1].

Definition 1.22. Let V= (vn)n∈Nbe a decreasing sequence of strictly posi-

tive continuous weights on X.

i) V= (vn)n∈Nsatisfies condition (M, K) if, for every non relatively com-

pact subset Yof X,∀n∈N∃˜n∈Nsuch that

inf

x∈Y

v˜n(x)

vn(x)= 0.

ii) V= (vn)n∈Nsatisfies condition (M1,K) if, for every non relatively com-

pact subset Yof X, there is n∈Nsuch that

inf

x∈Y

vn(x)

v1(x)= 0.

iii) the family Vsatisfies the continuous domination property if every v∈

Vis dominated by a continuous element of V.

Proposition 1.23. (Bastin [1]) Let V= (vn)n∈Nbe a decreasing sequence

of strictly positive continuous weights on X. The following conditions are

equivalent:

i) V satisfies condition (M, K)and the continuous domination property,

ii) V satisfies condition (M1,K)and the continuous domination property,

iii) the space Xis locally compact and σ-compact.

Theorem 1.24. (Bierstedt, Bonet [6], Bastin [2]) Let Xbe σ-compact. In

the (LB)-case of VC(X)the following conditions are equivalent:

i) The sequence Vsatisfies condition (D),

ii) VC(X) = CV(X)holds algebraically and topologically.

It follows a collection of results for projective description in the case of (LF)-

spaces. The (LF)-spaces VC(X) and V0C(X) were defined by Bierstedt,

Bonet and investigated in [8]. The notation and the main results of this

article are given below.

For every n∈Nlet Vn= (vn,k)k∈Nbe an increasing sequence of strictly

positive continuous functions on X. Let Vdenote the sequence (Vn)n∈Nand

let us assume that

vn+1,k(x)≤vn,k(x)≤vn,k+1(x)

holds for all n, k ∈Nand for all x∈X. Define

CVn(X) := {f∈C(X); ∀k∈N:||f||n,k := sup

x∈X

vn,k(x)|f(x)|<∞},

C(Vn)0(X) := {f∈C(X); ∀k∈N∀ε > 0∃K⊂Xcompact :

vn,k(x)|f(x)| ≤ ε∀x∈X\K}.

For each n∈Nwe obtain that CVn(X) (resp. C(Vn)0(X)) is continuously

included in CVn+1(X) (resp. C(Vn+1)0(X)). The weighted (LF)-spaces of

continuous functions are defined by

VC(X) := indnCVn(X) and V0C(X) := indnC(Vn)0(X).

Remark 1.25. As an (LF )-space VC(X) (resp. V0C(X)) is webbed and

ultrabornological.

This holds because CVn(X) (resp. C(Vn)0(X)) is webbed and ultrabornolog-

ical as a Fr´echet space for each n∈Nand because a countable inductive

limit of webbed or ultrabornological spaces is webbed or ultrabornological

(see theorem 1.2 and theorem 1.5).

In order to describe VC(X) and V0C(X) algebraically and topologically Bier-

stedt, Bonet [8] introduced the system Vof weights associated with V,

V:= {v∈C(X); v≥0 and ∀n∈N∃αn>0, k(n)∈N:v≤αnvn,k(n)}.

The projective hulls of the weighted inductive limits are defined as follows:

CV (X) := {f∈C(X); ∀v∈V:pv(f) := sup

x∈X

v(x)|f(x)|<∞},

CV 0(X) := {f∈C(X); ∀v∈V∀ε > 0∃K⊂Xcompact :

v(x)|f(x)| ≤ ε∀x∈X\K}.

One has VC(X)⊂CV (X) and V0C(X)⊂CV 0(X) with continuous inclu-

sions, and CV 0(X) and CV (X) are complete locally convex spaces.

The main results of [8] are that VC(X) = CV (X) holds algebraically if

and only if the sequence Vsatisfies condition (wQ), and that the (LF)-space

VC(X) is also complete if and only if the sequence Vsatisfies condition (wQ).

In the case of o-growth conditions it was proved that V0C(X) = CV 0(X)

is equivalent to Vsatisfying condition (Q), and that this is equivalent to

V0C(X) complete.

1.5 Known results of projective spectra of (DF )-spaces

Before we can investigate the structure of the (PLB)-spaces AC(X) and

A0C(X), we need some general results about projective spectra of (DF )-

spaces. All the following results go back to the work of Palamodov [36],

Retakh [38], Vogt [42],[43], and Wengenroth [45].

Definition 1.26. Aprojective spectrum Xis a sequence (Xn)n∈Nof linear

spaces (over the same field of real or complex numbers) and linear maps

ιn

m:Xm→Xnfor n≤m, satisfying

ιn

m◦ιm

k=ιn

kfor n≤m≤kand ιn

n= idXn.

Write X= (Xn, ιn

n+1)n∈N.

Definition 1.27. For X= (Xn, ιn

n+1)n∈Nset:

Proj0X:= {(xn)n∈N∈Y

n∈N

Xn;ιn

m(xm) = xn∀n≤m},

Proj1X:= Y

n∈N

Xn/B(X),

where

B(X) = {(an)n∈Y

n∈N

Xn;∃(bn)n∈Y

n∈N

Xnsuch that an=ιn

n+1bn+1 −bn}.

There is a natural exact sequence of linear spaces:

(∗) 0 −→ Proj0X,→Y

n∈N

−→ Y

n∈N

−→Proj1X −→ 0,

where σ: (xn)n∈N→(ιn

n+1xn+1 −xn)n∈Nand qis the quotient map.

In the case of a projective spectrum X= (Xn, ιn

n+1) of (LB)-spaces every Xn

has the form Xn=∪kXn,k where Xn,k is a Banach space with a norm ||·||n,k,

and Xncarries the locally convex inductive limit topology of the Xn,k.ιn

n+1

is assumed to be continuous. We put Bn,k := {x∈Xn,k;||x||n,k ≤1}and

assume that ∪k∈NBn,k =Xnand that (Bn,k)k∈Nis a fundamental sequence of

bounded sets in Xn. Let X= projnXn.ιn:X→Xndenotes the canonical

projection onto the n-th component. Xis called reduced if Xn=ιnXfor all

n∈N.Xis called a (DF S)-spectrum if for every kand mthere exists M

such that the inclusion Xk,m ,→Xk,M is compact. For a locally convex space

Xwe denote by U0(X) the filter basis of absolutely convex neighbourhoods of

0. Palamodov and Retakh investigated under which conditions Proj1X= 0,

i.e. the map σin the exact sequence (∗) is surjective. First Palamodov [36]

presented a sufficient condition:

Theorem 1.28. Let Xbe a projective spectrum and assume that each Xn

is endowed with a complete metrizable group topology such that the spectral

maps are continuous and

∀n∈N, U ∈ U0(Xn)∃m≥n∀µ≥m:ιn

mXm⊂ιn

µXµ+U.

Then Proj1X= 0.

The next result was given by Frerick and Wengenroth [22] and independently

by Braun and Vogt [19]. BD(X) denotes the set of Banach discs in a locally

convex space X.

Theorem 1.29. Let X= (Xn, ιn

m)be a projective spectrum consisting of

locally convex spaces and continuous linear maps such that

∀n∈N∃m≥n∀µ≥m∃B∈ BD(Xn)∀M∈ BD(Xm)

∃K∈ BD(Xmu); ιn

m(M)⊂ιn

µ(K) + B.

Then Proj1X= 0.

Palamodov gave a characterisation in the case of a projective spectrum of

Fr´echet spaces by showing that then the condition in theorem 1.28 is also

necessary. Compare also Wengenroth ([48], 3.2.8).

Theorem 1.30. For a projective spectrum Xconsisting of Fr´echet spaces

and continuous linear maps the following conditions are equivalent:

i) Proj1X= 0,

ii) ∀n∈N, U ∈ U0(xn)∃m≥n∀µ≥m:ιn

mXm⊂ιn

µXµ+U.

In the case of projective spectra of (LB)-spaces Retakh gave a necessary and

sufficient condition for Proj1X= 0. Compare also [48], 3.2.9.

Theorem 1.31. For a projective spectrum Xof (LB)-spaces, Proj1X= 0

holds if and only if there is a sequence of Banach discs Bn⊂Xnsuch that

i) ιn

mBm⊂Bnfor all n≤m,

ii) for every nthere is m≥nsuch that for each µ≥m

ιn

m(Xm)⊂ιn

µXµ+Bnholds.

Vogt (see [42], theorem 4.4, proposition 4.5, theorem 5.7) reformulated these

results and introduced condition (P).

Theorem 1.32. For a projective spectrum Xof (LB)-spaces, Proj1X= 0

holds if and only if the following holds:

(P) ∃(k(ν))ν∈N∀n∈N∃m≥n∀µ≥m∃S:

ιn

mXm⊂ιn

mXµ+S

ν=1

(ιν

n)−1Bν,k(ν).

Proposition 1.33. The following is necessary for (P):

∃(k(ν))ν∈N∀n∈N∃m≥n∀l, µ ∃L, S :

ιn

mBm,l ⊂S(ιn

µBµ,L +

ν=1

(ιν

n)−1Bν,k(ν)).

In [42], 5.7 Vogt showed the connection between the projective spectrum X

and the topological properties ultrabornological resp. barrelled for its pro-

jective limit.

Theorem 1.34. For a projective spectrum Xof (LB)-spaces the following

holds: If Proj1X= 0, then X= projnXnis ultrabornological (and hence

barrelled).

The following properties were defined in [43]:

Definition 1.35. For a projective spectrum Xlet

(P1)∃k∀n∃m∀µ, l ∃L, S :ιµ

mBm,l ⊂S(ιµ

nBn,k +Bµ,L),

(P2)∀n∃k, m ∀µ, l ∃L, S :ιµ

mBm,l ⊂S(ιµ

nBn,k +Bµ,L).

Of course (P1) is stronger than (P2). A weak variant of condition (P2) was

defined by Wengenroth [45].

Definition 1.36. Let X= (Xn, ιn

m) be a projective spectrum and B(Xn) the

family of all absolutely convex bounded sets. Then

(P3)∀n∈N∃m≥n∀µ≥m∃B∈ B(Xn)∀M∈ B(Xk)∃K∈

B(Xµ), S > 0 :

ιn

m(M)⊂S(ιn

µ(K) + B).

Vogt turned the conditions (P1) and (P2) into inequalities by means of duali-

sation. The following notation was used: Let jm

n:X0

n→X0

mbe the transpose

of ιn

mfor n≤m. For y∈X0

nset

||y||∗

n,k = sup{|y(x)|:||x||n,k ≤1}

with

||y||∗

n,k ≤ ||y||∗

n,k+1,

||y||∗

n,k ≥ ||jk+1

ny||∗

n+1,k.

For a reduced spectrum Xwe identify X0

nwith X∗

n:= jnX0

n⊂X0, where jn

is the transpose of ιn. Then X∗

n⊂X∗

n+1 holds for each n∈Nand we obtain

an imbedding spectrum of Fr´echet spaces. By X∗we denote the dual space

X0equipped with the inductive topology. X0

bdenotes X0equipped with the

strong topology. The map id : X∗→X0

bis continuous.

Definition 1.37.

(P∗

1)∃k∀n∃m∀µ, l ∃L, S > 0∀y∈X∗

n:||jm

ny||∗

m,l ≤S(||jµ

ny||∗

µ,L+||y||∗

n,k),

(P∗

2)∀n∃k, m ∀µ, l ∃L, S > 0∀y∈X0

n:||jm

ny||∗

m,l ≤S(||jµ

ny||∗

µ,L+||y||∗

n,k).

Condition (P1) and (P∗

1) are not equivalent in general. The same holds for

(P2) and (P∗

2). An example for this was given by Dierolf, Frerick, Mangino

and Wengenroth [20]. They constructed a projective spectrum of (LB)-

spaces of ”Moscatelli type” which satisfies the conditions (P∗

1) and (P∗

2), but

neither (P1) nor (P2). On the other hand, with duality theory and the bipolar

theorem it follows:

Remark 1.38. (Vogt [43]) For (DF S)-spectra the conditions (P∗

1) and (P∗

are equivalent to (P1) and (P2), respectively.

Theorem 1.39. (Vogt [43]) Let Xbe a (DF S)-spectrum. The following

implications hold:

i) (P1)⇒Proj1X= 0 ⇒(P2),

ii) (P∗

1)⇒Proj1X= 0 ⇒(P∗

2).

Now assume that Xis reduced and consider the inductive spectrum X∗. For

a reduced spectrum Vogt (see [42] corollary 5.10, theorem 5.11) showed:

Corollary 1.40. If Xis barrelled and Xis reduced, then

(P∗

2)∀n∃k, m ∀µ, l ∃L, S > 0∀y∈X0

||jm

ny||∗

m,l ≤S(||jµ

ny||∗

µ,L +||y||∗

n,k)

holds.

Theorem 1.41. If Xis reduced, the following implications hold:

Proj1X= 0 ⇒X∗is regular ⇒(P∗

2).

(P)⇔Proj1X= 0

⇓

Xultrabornological ⇒Xbarrelled

⇓

X0sequentially complete

(quasi complete)

⇔

⇒

Every bounded set in X0is contained

and bounded in some X∗

⇓ ⇓

Xbornological

⇓

bcomplete

⇒Xquasibarrelled

⇓

bsequentially complete

(quasi complete)

⇒

Every bounded set in X0

bis contained

and bounded in some X∗

⇓

X∗regular

⇓

(P∗

For a reduced spectrum Xthe implications between the properties discussed

in this section are collected in the previous scheme. All these implications

were shown by Vogt in [42] and [43].

Asking under which conditions the map

σ:Y

n∈N

Xn→Y

n∈N

Xn, σ((xn)n∈N) = (ιn

n+1xn+1 −xn)n∈N

in the exact sequence (∗) is not only surjective, but also every bounded set

in Qn∈NXnis contained in the image under σof a bounded set in Qn∈NXn

(”σlifts bounded sets”), we are led to the following definition.

Definition 1.42. (Bonet, Dierolf, Wengenroth [18]) Let B(Xn) be the system

of bounded sets of Xn, n ∈N. A projective spectrum X= (Xn, ιn

m) consisting

of locally convex spaces is said to be of strong P-type if

(Ps)∀n∈N∃Bn∈ B(Xn), m ≥n∀µ≥m, M ∈ B(Xm)∃K∈ B(Xµ) :

ιn

m(M)⊂ιn

µ(K) + Bn.

A similar condition was used by Wengenroth in the following theorem:

Theorem 1.43. (Wengenroth [48]) Let X= (Xn, ιn

m)be a locally convex

projective spectrum of locally complete spaces. Then the condition

(P˜s)∀n∃m≥n∀µ≥m∃B∈ B(Xn)∀M∈ B(Xm)∃K∈ B(Xµ)

ιn

m(M)⊂ιn

µ(K) + B

implies that σlifts bounded sets, that there are ˜

Bn∈ B(Xn)with ιn

m(˜

Bn)⊂˜

for M≥nand that ∀n∈N∃m≥n∀M∈ B(Xm)∃D∈ B(ProjX)

ιn

m(M)⊂ιn(D) + ˜

Bn.

Corollary 1.44. (Wengenroth [48]) Let X= (Xn, ιn

m)be a locally convex

projective spectrum of regular (LB)-spaces. Then σlifts bounded sets if and

only if the condition (P˜s)of 1.43 holds.

1.6 New results on the weighted (P LB)-spaces A0C(X)

and AC(X)

1.6.1 Structure of A0C(X)

Using the general results of Vogt and Wengenroth we now investigate the

structure of the weighted (PLB)-space A0C(X). In the case of the (LB)-

spaces (An)0C(X) the linear maps

ιn

m: (Am)0C(X)→(An)0C(X)

for n≤mcan be chosen as ιn

m= id(Am)0C(X)since (Am)0C(X)⊂(An)0C(X).

The projective spectrum is defined by

A0:= ((An)0C(X),id(An+1)0C(X))n∈N,

and its projective limit A0C(X) is

A0C(X) := projn(An)0C(X) = projnindkC(an,k)0(X).

Remark 1.45. The projective spectrum A0is reduced.

Proof: Let Cc(Rn) be the space of all continuous functions with compact

support and let f∈C(an,k)0(X). For each ε > 0 there exists a compact set

K⊂Xwith

an,k(x)|f(x)|< ε ∀x∈X\K.

One can choose a function ϕ∈Cc(Rn) with ϕ≡1 on Kand look at fϕ ∈

Cc(Rn). It follows that Cc(Rn) is dense in C(an,k)0(X) for each n, k ∈Nand

hence in each (An)0C(X) = indkC(an,k)0(X). Because Cc(Rn) is dense in

each step it is dense in the projective limit A0C(X).

Remark 1.46. In general A0C(X) is not a (DF S)-spectrum, because the

inclusions are not compact.

To see under which conditions Proj1A0= 0, i.e. the map σin the canonical

exact sequence

0−→ Proj0A0,→Y

n∈N

(An)0C(X)σ

−→ Y

n∈N

(An)0C(X)q

−→Proj1A0−→ 0

is surjective, we use condition (wQ) on A= ((an,k)k∈N)n∈N. Before we can

prove the main theorem of this chapter, which will show the connection

between condition (wQ), Proj1A0= 0 and A0C(X) ultrabornological resp.

barrelled, we need the following obvious result:

Lemma 1.47. Let Xbe locally compact and Z1, Z2⊂Xzero sets of con-

tinuous functions h1, h2on X. If Z1∩Z2=∅, then h=|h1|

|h1|+|h2|defines

a continuous function on Xwith values in [0,1] such that h|Z1≡0and

h|Z2≡1.

Theorem 1.48. The following conditions are equivalent:

i) The sequence A= ((an,k)k∈N)n∈Nsatisfies condition (wQ),

ii) Proj1A0= 0,

iii) the (P LB)-space A0C(X)is ultrabornological,

iv) the (PLB)-space A0C(X)is barrelled.

Proof. We will prove the theorem in the following way

i)1.)

=⇒ii)2.)

=⇒iii)3.)

=⇒iv)4.)

=⇒i).

For 1.) we apply theorem 1.32. In the case of the weighted (P LB)-spaces

A0C(X) condition (P) looks as follows:

∃(k(ν))ν∈N∀n∈N∃m≥n∀µ≥m∃S > 0 :

(Am)0C(X)⊂(Aµ)0C(X) + S∩n

ν=1 (Bν,k(ν))0.

Let A= ((an,k)k∈N)n∈Nsatisfy condition (wQ), and therefore (wQ∗). Us-

ing condition (wQ∗) one can find a sequence (k(ν))ν∈Nwhich satisfies (P).

For given nselect maccording to (wQ∗), and for given µ≥mand f∈

(Am)0C(X) one can find lsuch that f∈C(am,l)0(X). For µ, l select L, S > 0

as in (wQ∗). Now define the sets

K:= {x∈X;am,l(x)|f(x)|>1

2S},

X1:= {x∈X;aµ,L(x)<2Sam,l(x)},

X2:= {x∈X;am,l(x)|f(x)|<1

S} ∩ {x∈X;aµ,L(x)> Sam,l(x)}.

The set Kis relatively compact and open in X. We claim that

X= (X1∪K)∪X2

holds. To show this, let x∈Xbe given. The following cases are possible:

I) am,l(x)|f(x)|>1

2Sor II) am,l(x)|f(x)| ≤ 1

2S<1

In the first case x∈K. If the second case is true one has to evaluate if

a) aµ,L(x)<2Sam,l(x) or b) aµ,L(x)≥2Sam,l(x)> Sam,l(x).

From a) it follows that x∈X1. If b) is true, then x∈X2holds.

Now define Z1:= X\(X1∪K) and Z2:= X\X2. For Z1and Z2we

obtain

Z1∩Z2=X\(X1∪K)∩(X\X2) = X\(X1∪K∪X2) = ∅.

To apply lemma 1.47 one has to show that Z1and Z2are zero sets of

continuous functions h1, h2on X. The set Z1can be written as Z1=

(X\X1)∩(X\K). First we show the existence of suitable continuous

functions g1, g2such that X\X1and X\Kare zero sets. Define the con-

tinuous functions g1and g2on Xby

g1(x) := 0 for aµ,L(x)−2Sam,l(x)>0

aµ,L(x)−2Sam,l(x) elsewhere,

g2(x) := 0 for am,l(x)|f(x)| − 1

2S<0

am,l(x)|f(x)| − 1

2Selsewhere.

X\X1is the zero set of g1,X\Kis the zero set of g2and finally Z1is the

zero set of the function h1:= max(g1, g2). The case of Z2can be treated in

a similar way.

By lemma 1.47 there exists a function h∈C(X, [0,1]) such that h|Z1≡0

and h|Z2≡1. Write the given function f∈C(am,l)0(X) as f=hf + (1 −

h)f=: f1+f2. We have to show that f1∈C(aµ,L)0(X) and therefore

f1∈(Aµ)0C(X) and f2∈S(Bν,k(ν))0for ν= 1, ..., n. If x /∈X1∪Kthen

f1(x) = h(x)f(x) = 0. If x∈X1∪Kit follows that |f1(x)| ≤ |f(x)|and for

x∈X1

aµ,L(x)|f1(x)| ≤ aµ,L(x)|f(x)| ≤ 2Sam,l(x)|f(x)|

holds. Since Kis compact,

sup

x∈X

aµ,L(x)|f1(x)|<∞.

Collecting these facts we obtain that for x∈X1∪Kthere exists ˜

S > 0

such that aµ,L|f1| ≤ ˜

Sam,l|f|. From f∈C(am,l)0(X) it follows that f1∈

C(aµ,L)0(X). If x /∈X2

f2(x) = f(x)(1 −h(x)) = 0.

On the other hand, if x∈X2we have |f2| ≤ |f|and by the definition of X2

(∗∗)1

am,l(x)>S

aµ,L(x).

With condition (wQ∗) and (∗∗) we obtain that

aν,k(ν)(x)≤Sam,l(x)

holds for ν= 1, ..., n, and then

aν,k(ν)(x)|f2(x)| ≤ aν,k(ν)(x)|f(x)| ≤ Sam,l(x)|f(x)|

holds for ν= 1, ..., n. It follows that f2∈C(aν,k(ν))0(X) for ν= 1, ..., n. From

x∈X2we also can conclude that am,l(x)|f(x)|<1

S, which is equivalent to

Sam,l(x)|f(x)|<1. Then

aν,k(ν)(x)|f2(x)| ≤ Sam,l(x)|f(x)|<1

holds for ν= 1, ..., n and therefore f2∈(Bν,k(ν))0for ν= 1, ..., n. Finally

condition (P) and therefore Proj1A0= 0 follows.

2.) follows immediately from theorem 1.34 and 3.) holds in general. Since

the projective spectrum A0is reduced we can apply corollary 1.40 to show

4.):

(P∗

2)∀n∃m≥n, k ∀µ≥m, l ∃L, S ∀ϕ∈(An)0C(X)0:

||ϕ||∗

m,l ≤S(||ϕ||∗

µ,L +||ϕ||∗

n,k),

where ||ϕ||∗

m,l := sup{| < ϕ, f > |;f∈(Bm,l)0}.Since (Bm,l)0⊂(Bn,l)0

for m≥n, the sup is finite. We show that condition (wQ) follows for the

sequence A. The quantifiers are the same. We fix x∈X. The measure

δx: (An)0C(X)→C, δx(f) := f(x), is continuous because the topology of

this weighted (LB)-space is finer than the compact-open topology. Now

||δx||∗

m,l = sup{|f(x)|;f∈(Bm,l)0} ≤ 1

am,l(x)

clearly holds since |f| ≤ 1

am,l on Xfor each f∈(Bm,l)0. On the other hand,

select ϕ∈C(X) with compact support, 0 ≤ϕ≤1 and ϕ(x) = 1. Clearly

f0=ϕ

am,l belongs to (Bm,l)0and

am,l(x)=f0(x)≤sup{|f(x)|;f∈(Bm,l)0}=||δx||∗

m,l.

Thus, we have proved that ||δx||∗

m,l =1

am,l(x). Therefore

am,l(x)≤S(1

aµ,L(x)+1

an,k(x))≤2Smax( 1

aµ,L(x),1

an,k(x))

which is condition (wQ).

For the next theorem we need the assumption that the space (An)0C(X) is

complete. This completeness was characterised by Bierstedt, Bonet, Sum-

mers [14] (see theorem 1.20).

Theorem 1.49. Let (An)0C(X)be complete for each n∈N. The projective

spectrum A0is of strong P-type if and only if the sequence A= ((an,k)k∈N)n∈N

satisfies condition (Q).

To prove this theorem we need some technical tools. First we present a

partition of a continuous functions. The idea of this goes back to Ernst,

Schnettler [21].

Proposition 1.50. Let u, v ∈C(X)be strictly positive functions. If f∈

C(X)satisfies |f| ≤ max(u, v), there exist g1, g2∈C(X)with |g1| ≤ uand

|g2| ≤ von Xsuch that |f|=g1+g2.

Proof. Define

A:= {x∈X;|f(x)| ≤ u(x)},

B:= {x∈X;|f(x)|> u(x)},

and g1, g2∈C(X) by g1(x) := min(|f(x)|, u(x)) and g2(x) := |f(x)| − g1(x).

Of course, |f|=g1+g2holds. If x∈Athen g1(x) = |f(x)| ≤ u(x) and

g2(x) = 0. For x∈Bwe obtain g1(x) = u(x) and g2(x) = |f(x)| − u(x).

With the definition of the set Band the assumption that |f| ≤ max(u, v) it

follows that v(x)≥ |f(x)| ≥ |f(x)| − u(x) = g2(x). We have 0 ≤g1≤uand

0≤g2≤von X.

Lemma 1.51. Let A0be of strong P-type. Condition (Ps)(see definition

1.42) can be written as follows: ∀n∃m, k ∀µ, l, ε > 0∃L, S > 0 :

(Bm,l)0⊂S(Bµ,L)0+ε(Bn,k)0.

Then ∀n∃m, k ∀µ, l, ε > 0∃L, S > 0∀ϕ∈(An)0C(X)∗:

||ϕ||∗

m,l ≤S||ϕ||∗

µ,L +ε||ϕ||∗

n,k

holds.

Proof. Choose ϕ∈(An)0C(X)∗and let ||ϕ||∗

m,l := sup{|ϕ(f)|;f∈(Bm,l)0} ≤

1. Write fas f=Sf1+εf2with f1∈(Bµ,L)0and f2∈(Bn,k)0.

|ϕ(f)| ≤ S|ϕ(f1)|+ε|ϕ(f2)|

≤S||ϕ||∗

µ,L +ε||ϕ||∗

n,k

holds for all f∈(Bm,l)0. Then the same inequality holds for the sup, and

one gets

||ϕ||∗

m,l ≤S||ϕ||∗

µ,L +ε||ϕ||∗

n,k.

Proof of 1.49. Let the projective spectrum A0be of strong P-type. We apply

lemma 1.51 and get: ∀n∃m, k ∀µ, l, ˜ε > 0∃L, ˜

S > 0 such that

||ϕ||∗

m,l ≤˜

S||ϕ||∗

µ,L + ˜ε||ϕ||∗

n,k.

Let n, m, k, µ, l be as before and put ε= 2˜ε. Let Lbe as above and put

S= 2 ˜

S. Fix x0∈Xand define δx0: (An)0C(X)→Cas in the proof of

theorem 1.48; recall that from 4.) in the proof of theorem 1.48

||δx0||∗

m,l = sup{|f(x0)|;f∈(Bm,l)0}=1

am,l(x0).

It follows that

am,l(x0)=||δx0||∗

m,l ≤˜

S||δx0||∗

µ,L + ˜ε||δx0||∗

n,k

=˜

aµ,L(x0)+ ˜ε1

an,k(x0).

We claim that 1

am,l(x0)≤max(S1

aµ,L(x0), ε 1

an,k(x0)),

which is exactly condition (Q). If not, then

am,l(x0)=1

am,l(x0)+1

am,l(x0)

2S1

aµ,L(x0)+1

2ε1

an,k(x0),

which implies 1

am,l(x0)>˜

aµ,L(x0)+ ˜ε1

an,k(x0),

a contradiction to the inequality proved above.

Now let condition (Q) be given. Take f∈Bm,l; then

|f| ≤ 1

am,l

≤max( S

aµ,l

,ε

an,k

)

holds on X. By proposition 1.50 there exist f1, f2∈C(X) with |f1| ≤

aµ,L ,|f2| ≤ ε

an,k and |f|=f1+f2. It follows that f1∈SBµ,L and f2∈εBn,k.

1.6.2 Structure of AC(X)

In this chapter we investigate the structure of the space AC(X) which is

defined by

AC(X) := projnAnC(X) = projnindkCan,k(X).

Similar to the case of o-growth conditions the linear maps

ιn

m:AmC(X)→ AnC(X)

for n≤mcan be chosen as ιn

m= idAmC(X)since AmC(X)⊂ AnC(X). The

projective spectrum is defined by

A:= (AnC(X),idAn+1C(X))n∈N.

Theorem 1.52. Proj1A= 0 if and only if A= ((an,k)∈N)n∈Nsatisfies con-

dition (Q).

Proof. Without loss of generality we can assume that for each n∈Nthe

system (Bn,k)k∈Nis a fundamental system of bounded sets in AnC(X). First

let Proj1A= 0. With theorem 1.31 it follows that for each n∈Nthere exists

a bounded absolutely convex Bnin AnC(X) such that Bn+1 ⊂Bnfor each

n∈Nand

(∗∗)∀n∃m≥n∀µ≥m:AmC(X)⊂ AµC(X) + Bn.

Since Bnis bounded in AnC(X) and since AnC(X) is a regular inductive

limit (see [14]), we can select k(n)∈Nsuch that Bn⊂Bn,k(n). Now we have

to show that A= ((an,k)∈N)n∈Nsatisfies condition (Q). For given n, select

mas in (∗∗) and k=k(n) with Bn⊂Bn,k(n). Fix µ≥m, l, ε > 0; w.l.o.g.

ε≤1. For the function f:= 1

εam,l ,f∈ AmC(X) holds, and by (∗∗) one

can write f=g+hwith g∈ AµC(X) and h∈Bn. For g∈ AµC(X) there

exist L∈Nand S > 0 with aµ,L|g| ≤ Son X. For h∈Bnit follows that

h∈Bn,k(n)and then an,k|h| ≤ 1, and therefore |h| ≤ 1

an,k holds on X. It

follows that

εam,l

=|f| ≤ |g|+|h| ≤ S

aµ,L

an,k

⇒1

am,l

≤Sε

2aµ,L

+ε

2an,k

≤max( S

aµ,L

,ε

an,k

)

holds on X, which is exactly (Q).

In the other direction we show that condition (Q) even implies that the

map σin the exact sequence lifts bounded sets. Then Proj1A= 0 fol-

lows immediately from theorem 1.29. Let condition (Q) be satisfied, i.e.

∀n∃m, k ∀µ, l, ε > 0∃L, S > 0 :

am,l

≤max( S

aµ,L

,ε

an,k

)

holds on X. Take f∈Bm,l; then

|f| ≤ 1

am,l

≤max( S

aµ,l

,ε

an,k

)

holds on X. By proposition 1.50 there exist f1, f2∈C(X) with |f1| ≤

aµ,L ,|f2| ≤ ε

an,k and |f|=f1+f2. It follows that f1∈SBµ,L and f2∈εBn,k,

which is condition (Ps) in the case of the space AC(X) (compare lemma

1.51).

Remark 1.53. We have even shown that AC(X) has Proj1A= 0 if and

only if the projective spectrum Ais of strong P-type.

Remark 1.54. With theorem 1.52 above and theorem 1.34, a general result

for projective spectra of (LB)-spaces, we get the following inclusions for the

(PLB)-space AC(X):

(Q)⇒Proj1A= 0 ⇒ AC(X) is ultrabornological (and hence barrelled).

But in general the (PLB)-space AC(X) is not reduced, i.e. we cannot use

the general theory of (DF )-spaces to conclude that from AC(X) barrelled

or ultrabornological it follows that the sequence Asatisfies condition (wQ)

or (Q). It is unknown if then condition (Q) must be satisfied. The fact

that from AC(X) barrelled it follows that the sequence Asatisfies at least

condition (wQ) is indeed right. But to show this, we need the inductive

description which will be introduced in the next section.

1.7 Inductive description

From now on through the whole section 1.7 let the space Xbe locally compact

and σ-compact.

1.7.1 Inductive description for Fr´echet spaces

First we investigate inductive description for an increasing sequence A=

(an)n∈Nof strictly positive continuous functions (weights). For

Can(X) := {f∈C(X); ||f||n:= sup

x∈X

an(x)|f(x)|<∞},

the space CA(X) = projnCan(x) is a Fr´echet space. Define

A:= {a∈C(X); a≥0,∀n∃αn>0 : a≥αnan},

Ca(X) := {f∈C(X); ||f||a:= sup

x∈x

a(x)|f(x)|<∞}

and the corresponding space

CA(X) := inda∈ACa(X).

Theorem 1.55.

CA(X) = CA(X)

holds algebraically, and the canonical mapping CA(X)→CA(X)is contin-

uous.

Proof. Let f∈CA(X). There exists a∈A, a ≥αnanfor each n∈N, with

f∈Ca(X). It follows that

||f||n= sup

x∈X

an(x)|f(x)| ≤ 1

αn

sup

x∈X

a(x)|f(x)|=1

αn

||f||a,

which means that f∈Can(X) for each n∈N, hence f∈CA(X).

Now let g∈CA(X). Then for each n∈Nthere exists αn∈R+with

an|g| ≤ αnfor all n∈N. With the inequality above we obtain |g| ≤ αn

anon

Xfor each n∈N. It follows that |g| ≤ infn∈Nαn

anon X. Define the sequence

(wn)n∈Nby wn:= 1

an. In the notation of section 1.4.3 (but Vreplaced by W)

we obtain w:= infn∈Nαn

an∈W. With a result of Bierstedt, Meise, Summers

([14], 0.2) it follows that there exists ˜w∈Wwith ˜w(x)>0 for all x∈X

and w= infn∈Nαn

an≤˜w. Define b:= 1

˜w. Then |g| ≤ ˜w=1

b, and therefore

|g|b≤1 holds. We still have to show that b∈Aholds. From ˜w∈Wit

follows that ∀n∈N∃βn>0 such that 1

b= ˜w≤infnβnwn= inf βn1

an.

Then b≥supn1

βnan, and this means that b∈Aand g∈CA(X).

1.7.2 Inductive description in the (PLB)-case

Now we investigate the (P LB)-case. Again we take a double sequence A=

((an,k)k∈N)n∈Nof strictly positive weights with

an,k+1(x)≤an,k(x)≤an+1,k(x)∀n, k ∈N,∀x∈X,

and define

A:= {a∈C(X); a≥0,∀n∈N∃αn, k(n) : a≥sup

αnan,k(n)}.

Again we have vn,k =1

an,k where vn,k ∈Vis a weight as in [8] with

V={v∈C(X); v > 0,∀n∈N∃αn, k(n) : v≤inf

nαnvn,k(n)}.

Then A={1

v;v∈V}. Define

Ca(X) := {f∈C(X); ||f||a:= sup

x∈X

a(x)|f(x)|<∞}

and

CA(X) := inda∈ACa(X).

Since Xis σ-compact, one can restrict the attention to the positive elements

v∈V(see [14]) and hence to the positive elements a∈A. Each Ca(X) is a

Banach space, a∈A, and hence CA(X) is ultrabornological.

Theorem 1.56.

CA(X) = AC(X)

holds algebraically, and the canonical mapping CA(X)→ AC(X)is contin-

uous.

Proof. Let f∈CA(X), i.e. there exists a∈Asuch that f∈Ca(X). For

each n∈Nthere are αn>0 and k(n)∈Nsuch that a≥αnan,k(n). It follows

that

||f||n,k(n)= sup

x∈X

an,k(n)(x)|f(x)| ≤ 1

αn

sup

x∈X

a(x)|f(x)| ≤ 1

αn

||f||a<∞,

which means f∈Can,k(n)(X) for each n∈N, hence f∈ AnC(X) for each

n∈Nand therefore f∈ AC(X). The above inequality shows that Ca(X)

is continuously injected in Can,k(n)(X) for each n∈N. It follows that the

mapping CA(X)→ AnC(X) is continuous for each n∈N, which proves the

last assertion of the theorem.

Now let f∈ AC(X), i.e. for each n∈Nthere exists k(n)∈Nsuch that

f∈Can,k(n)(X), i.e. for each n∈Nthere exist k(n)∈N, bn>0 such that

an,k|f| ≤ bn⇒ |f| ≤ bn1

an,k for all n∈N. Define w:= infnbn1

an,k(n). One has

w∈V, and with [14] there exists w∈Vwith w(x)>0 for all x∈Xand

w≤w. Define a:= 1

w∈C(X). Then a|f| ≤ 1 because |f| ≤ w≤w. It

follows that a∈Aand f∈CA(X).

Corollary 1.57. CA(X)and AC(X)have the same bounded sets, and the

inductive limit CA(X)is regular.

Proof. After theorem 1.56 it suffices to fix a bounded set B⊂ AC(X) and

to show that there exists a∈Asuch that Bis contained and bounded in

Ca(X). Bis bounded in AnC(X) for each n∈N. Since AnC(X) is a regular

inductive limit, there exist k(n), mn>0 such that B⊂mnBn,k(n). Then for

each f∈B

an,k(n)|f| ≤ mn

holds. Define w:= infnmn1

an,k(n). With the same arguments as above we

obtain a∈A, a > 0, such that a|f| ≤ 1 for each f∈B, which means that B

is contained in Ca(X) and bounded there.

Theorem 1.58. Let the sequence Asatisfy condition (Q). Then

CA(X) = AC(X)

holds topologically.

Proof. If condition (Q) is satisfied, with theorem 1.52 it follows that Proj1A=

0. Hence AC(X) is ultrabornological (theorem 1.34) and webbed (corollary

1.3). The space CA(X) is ultrabornological and webbed, since CA(X) is the

ultrabornological space associated with AC(X) by corollary 1.57 and [26],

13.3.3. Define id : CA(X)→ AC(X), which is a continuous embedding.

With de Wilde’s [47] closed graph theorem it follows that id−1:AC(X)→

CA(X) is continuous, too. So we have a topological isomorphism between

the spaces AC(X) and CA(X).

Theorem 1.59. Let (An)0C(X)be complete for each n∈N. If AC(X)is

barrelled, then A0C(X)is barrelled.

Proof. Let T0be a barrel in A0C(X), i.e. T0is absolutely convex, closed and

absorbant. Define

T:= {f∈ AC(X); ϕf ∈T0∀ϕ∈Cc(X),0≤ϕ≤1}.

We show that Tis a barrel in AC(X): Choose f, g ∈T, λ, µ ∈IK =Ror C

such that |λ|+|µ| ≤ 1. Since

ϕ(λf +µg) = λ(ϕf) + µ(ϕg)∈T0,

Tis absolutely convex.

Now let (fi)i∈N⊂Twith fi→f∈ AC(X) in AC(X). For ϕ∈Cc(X)

with 0 ≤ϕ≤1 we claim that ϕfi→ϕf in A0C(X). This is equivalent to

ϕfi→ϕf in (An)0C(X) for each n∈N. (An)0C(X) carries the topology

induced by its projective hull, say, C(An)0(X) for An= the “V“ associated

with (vn,k)k∈N. But for each v∈An:

pv(ϕ(fi−f)) = sup

x∈X

v(x)ϕ(x)|fi(x)−f(x)| ≤ pv(fi−f)→0

since fi→fin CAn(X) follows from fi→fin AnC(X). Since the barrel

T0is closed, ϕfi∈T0for each i∈Nnow implies ϕf ∈T0, and hence f∈T.

We have proved that Tis closed.

For f∈ AC(X) define

Bf:= {ϕf;ϕ∈Cc(X),0≤ϕ≤1}.

Bfis bounded in A0C(X) because for each n∈Nthere exist k(n)∈N, bn>0

such that an,k(n)|f| ≤ bnon Xand then

an,k(n)|ϕf| ≤ an,k(n)|f| ≤ bnon X,

which means that Bfis bounded in C(an,k(n))0(X), hence in (An)0C(X) and

then finally in A0C(X). After the assumption that (An)0C(X) is complete

(hence locally complete), there exists a Banach disc Bwith Bf⊂B. Since by

[37], 3.2.7, a barrel absorbs Banach discs, there is β > 0 such that Bf⊂βT0.

Then 1

βϕf ∈T0∀ϕ∈Cc(X) with 0 ≤ϕ≤1, hence 1

βf∈Tor f∈βT .

It follows that Tis absorbant. This finishes the proof that Tis a barrel in

AC(X).

Since AC(X) is barrelled, Tis a 0-neighbourhood, i.e. there exists W∈

U0(AC(X)) with W⊂T. After our hypothesis An= (an,k)k∈Nis regularly

decreasing for each n∈N, hence An= (an,k)k∈Nsatisfies condition (D),

and thus AnC(X) = CAn(X) holds topologically for each n∈N. Now the

0-neighbourhood Win AC(X) can be taken of the form

W={f∈ AC(X); sup

x∈X

an(x)|f(x)| ≤ 1}

with an∈Anfor some n∈N. Define

V:= {g∈ A0C(X); sup

x∈X

an(x)|g(x)| ≤ 1}.

{g∈(An)0C(X); supx∈Xan(x)|f(x)| ≤ 1}is a 0-neighbourhood in the space

(An)0C(X)⊂C(An)0(X), hence Vis a 0-neighbourhood in A0C(X), and

if we show V⊂T0, then T0is a 0-neighbourhood, too, which proves that

A0C(X) is barrelled. Let g∈V. Then g∈W⊂Tand hence ϕg ∈T0

for each ϕ∈Cc(X) with 0 ≤ϕ≤1. For each compact set K⊂Xlet

ϕK∈Cc(X) satisfy ϕK(x) = 1 for each x∈Kand 0 ≤ϕK≤1 on X.

Fix n∈Nand consider (ϕKg)Kin (An)0C(X). By using that (An)0C(X)

carries the topology induced by C(An)0(X), one easily sees that ϕKg→gin

(An)0C(X). Hence one has ϕKg→gin A0C(X), too, which yields g∈T0

since T0is closed in A0C(X).

Even without the hypothesis that (An)0C(X) is complete for each n∈N, a

modification of the proof of theorem 1.59 serves to show.

Corollary 1.60. If AC(X)is barrelled, then A0C(X)is quasibarrelled.

Proof. Let T0be a bonivorous barrel in A0C(X). Again define

T:= {f∈ AC(X); ϕf ∈T0∀ϕ∈Cc(X),0≤ϕ≤1}

and show that Tis a barrel in AC(X). That Tis absolutely convex and

closed follows exactly in the same way as in the proof of theorem 1.59. But

to show that Tis absorbant we now proceed as follows: For f∈ AC(X)

again define

Bf:= {ϕf;ϕ∈Cc(X),0≤ϕ≤1}.

As in the proof of 1.59, it is clear that Bfis bounded in A0C(X). Since here

T0is bornivorous, it follows that Bfis absorbed by T0. From this point on

the rest of the proof follows along the lines of the end of the proof of 1.59.

Remark 1.61. Let the following conditions be satisfied:

i) The sequence Asatisfies condition (Q),

ii) Proj1A= 0,

iii) AC(X) is barrelled,

iv) A0C(X) is quasibarrelled,

v) condition (P∗

2) is satisfied,

vi) the sequence Asatisfies condition (wQ).

Then the implications

i)⇒ii)⇒iii)⇒iv)⇒v)⇒vi)

hold.

It is an open question if iv)⇒ii) holds.

Proof. In theorem 1.52 we proved that condition (Q) for the sequence Ais

equivalent to Proj1A= 0. Hence it follows that AC(X) is barrelled (1.34).

In this section (corollary 1.60) we proved that AC(X) barrelled implies that

A0C(X) is quasibarrelled. Vogt proved in the general case of reduced spectra

of (LB)-spaces that the space Xis barrelled if and only if it is quasibarrelled

([43], 3.1). With corollary 1.40 it follows that condition (P∗

2) is satisfied,

which is equivalent to condition (wQ) (1.48).

1.8 Comparison of the (P LB)- and the (LF )-space

Now we want to describe under which conditions the (PLB)-spaces AC(X)

and A0C(X) are equal to the (LF )-spaces VC(X) and V0C(X), respectively.

This cannot be true in general as the following example shows for the case

of AC(X) and VC(X).

Example 1.62. First we define a sequence of weights on X:= N×Nby

an,k(i, j) := j−kif i≤n

0 otherwise , n, k ∈N.

A= ((an,k)k∈N)n∈Nis decreasing in kand increasing in n. Now define f:

N×N→R, f(i, j) := ji+1/2for each (i, j)∈N×N. Fix n, and select

k(n) = n+ 1. If i≤n, we have

|f(i, j)|=ji+1/2≤jn+1

for each j, which means that f∈Can,k(n)(X), hence f∈ AnC(X). Then we

get f∈ AC(X).

Suppose that f∈ VC(X).We find ksuch that for each nthere is Cn>0

with |f(i, j)| ≤ Cnjkfor i= 1, ..., n. Select n:= kand i:= kto conclude

jn+1/2=ji+1/2=|f(i, j)| ≤ Cnjk=Cnjn

for each j. This implies j1/2≤Cnfor each j, which is a contradiction.

Lemma 1.63. VC(X)⊂ AC(X)and V0C(X)⊂ A0C(X)holds in general

with continuous inclusions.

Proof. One can write VC(X) = indkprojnCan,k(X). Let f∈ VC(X). Thus

there exists k∈Nsuch that ||f||n,k <+∞for all n∈N. This implies

that for each n∈Nthere exists k∈Nsuch that ||f||n,k <+∞, hence

f∈ AnC(X) for each n∈Nand therefore f∈ AC(X). A similar argument

gives V0C(X)⊂ A0C(X). The argument for the continuous inclusions is as

follows:

projnCan,k(x)→Can,k(X)

is continuous for each n, k ∈N. Then

projnCan,k(x)→indkCan,k(X) = AnC(X)

is continuous for each n∈N, and thus

VC(X) = indkprojnCan,k(X)→ AnC(X)

must be continuous for each n∈N, whence the continuity of

VC(X)→ AC(X).

Next we introduce a condition which was used by Vogt [41] as a characteri-

sation for Fr´echet spaces between which all continuous linear mappings are

bounded.

Definition 1.64. Let A= ((am,l)l∈N)n∈Nbe a sequence of weights. Asatis-

fies condition (B) if ∀k(n)∃l∈N∀m∈N∃˜n∈N, c > 0:

am,l ≤cmax

1≤n≤˜nan,k(n).

Theorem 1.65. AC(X) = VC(X)holds algebraically if and only if the

sequence Asatisfies condition (B).

Proof. Let Asatisfy condition (B). VC(X)⊂ AC(X) holds by 1.63. To

show the other inclusion, choose f∈ AC(X). After the definition of AC(X)

it follows that: ∀n∈N∃k(n)∈N, bn>0 :

an,k(n)(x)|f(x)| ≤ bn∀x∈X.

For given (k(n))n∈Nwe can apply condition (B) to find lsuch that for each

mthere exist ˜n, c > 0 with

am,l ≤cmax

1≤n≤˜nan,k(n).

We claim that f∈CVl(X), i.e. f∈Cam,lC(X) for each m∈N. Indeed, for

given m, one can select ˜n:= ˜n(m) and c:= cm>0 as in condition (B) and

then for each x∈X:

am,l(x)|f(x)| ≤ cmax

1≤n≤˜nan,k(n)(x)|f(x)| ≤ cmax

1≤n≤˜nbn<∞,

and supx∈Xam,l(x)|f(x)|<∞ ∀ m∈N, hence f∈CVl(X)⊂ VC(X).

In the other direction, for given (k(n))n∈Nconsider the space

F:= ∩n∈NCan,k(n)(X)

={f∈C(X); sup

x∈X

an,k(n)(x)|f(x)|=: ||f||n<∞ ∀ n∈N}.

Clearly F⊂ AC(X). Observe that the norms || · ||ndo not satisfy

|| · ||n≤ || · ||n+1

in general. The space Fwith the norms pn(f) := max1≤ν≤n||f(x)||νis a

Fr´echet space because Fis continuously injected in AC(X), which has a

topology finer than the compact-open topology. By the assumption AC(X) =

VC(X) we get F⊂ VC(X) = indkCVk(X), which is an (LF)-space. Mor-

ever the inclusion has closed graph because VC(X) is continuously included

in AC(X). By Grothendieck‘s factorisation theorem there is lsuch that

F⊂CVl(X), and the inclusion is continuous. This implies ∀m∃˜n, c > 0

such that, for each g∈F,

(∗∗∗) sup

x∈X

am,l(x)|g(x)| ≤ cmax

1≤n≤˜nsup

x∈X

an,k(n)(x)|g(x)|.

Suppose ∃x0∈Xwith

am,l(x0)> α > c max

1≤n≤˜nan,k(n)(x0).

By continuity one can find a compact neighbourhood W0of x0with

cmax

1≤n≤˜nan,k(n)(x)≤α∀x∈W0.

Select ϕ∈C(X, [0,1]) with suppϕ⊂W0and ϕ(x0) = 1. Clearly ϕ∈F. We

apply (∗∗∗) to conclude

am,l(x0) = am,l(x0)|ϕ(x0)| ≤ sup

x∈X

am,l(x)|ϕ(x)|

≤cmax

1≤n≤˜nsup

x∈X

an,k(n)(x)|ϕ(x)| ≤ cmax

1≤n≤˜nsup

x∈W0

an,k(n)(x)

≤α.

This is a contradiction. Hence condition (B) holds.

Theorem 1.66. If the sequence Asatisfies condition (B), then the space

A0C(X)equals V0C(X)algebraically. If each (An)0(X)is complete, which

is equivalent to An= (an,k)k∈Nregularly decreasing for each n∈N, then the

converse is also true.

Proof. First we show that from condition (B) it follows that A0C(X) =

V0C(X) holds algebraically. V0C(X)⊂ A0C(X) holds in general (see 1.63).

Now let the sequence Asatisfy condition (B). Choose f∈ A0C(X). Then

after the definition of A0C(X) it follows that: ∀n∈N∃k(n)∈N∀ε >

0∃K(ε)⊂Xcompact:

(+) an,k(n)(x)|f(x)| ≤ ε∀x∈X\K(ε).

For given (k(n))n∈Nwe can apply condition (B) to find lsuch that for each

mthere exist ˜n= ˜n(m), c =cm>0 :

am,l ≤cmax

1≤n≤˜nan,k(n).

We claim that f∈C(Vl)0(X), i.e. f∈C(am,l)0(X) for each m∈N. Let

ε > 0 be given. For fixed n∈N,1≤n≤˜n, by (+) there exists k(n) such

that for ˜ε:= ε

cthere exists a compact set Kn⊂Xwith

an,k(n)(x)|f(x)| ≤ ˜ε∀x∈X\Kn.

Then

am,l(x)|f(x)| ≤ cmax

1≤n≤˜nan,k(n)(x)|f(x)| ≤ c˜ε=ε

for all x∈X\Kwith K:= ∪˜n

ν=1Kn, which proves our claim.

Now let all (An)0C(X) be complete and let A0C(X) = V0C(X) hold al-

gebraically. We have to show condition (B). Similarly to the proof of 1.65,

for given (k(n))n∈Nwe define F0:= ∩nC(an,k(n))0(X) and use the same ar-

guments to conclude that condition (B) is satisfied.

Corollary 1.67. If all (An)0C(X)are complete, then A0C(X) = V0C(X)

holds algebraically and topologically if and only if the sequence Asatisfies the

conditions (B)and (wQ).

Proof. When A0C(X) = V0C(X) holds topologically, then the space A0C(X)

is ultrabornological as an (LF )-space. With theorem 1.48 it follows that the

sequence Asatisfies condition (wQ), and condition (B) follows from 1.66.

Now let Asatisfy condition (wQ). With theorem 1.48 it follows that A0C(X)

is ultrabornological. As an (LF)-space V0C(X) is webbed. Define id :

V0C(X)→ A0C(X), which is a continuous embedding. With de Wilde’s [47]

closed graph theorem it follows that id−1:A0C(X)→ V0C(X) is continuous.

So we have a topological isomorphism between A0C(X) and V0C(X).

1.9 An example in the case of sequence spaces

Before we can illustrate the previous results with an example in the case

of sequence spaces we have to introduce K¨othe sequence spaces and some

of their properties. For the definitions and notations see Bierstedt, Meise,

Summers [15]. Some further results which are needed here were given by

Bierstedt, Bonet [5], [4].

Definition 1.68. Let A= (an)n∈Nbe an increasing sequence of strictly

positive functions on some index set I;Ais called a K¨othe matrix. In the

following example we chose I=Nand omit it from our notation.

We define the K¨othe echelon spaces of order p, 1≤p≤ ∞ or p= 0, as

follows:

λp(A) = {x∈IKN;||x||n:= (

∞

i=1

|xian(i)|p)1/p <∞ ∀ n∈N},1≤p≤ ∞,

λ∞(A) = {x∈IKN;||x||n:= sup

i∈N

|xi|an(i)<∞ ∀ n∈N},

λ0(A) = {x∈λ∞(A); lim

i→∞ xian(i) = 0 ∀n∈N}

with

λp(A) = projnlp(an(i)),1≤p≤ ∞,

and

λ0(A) = projnc0(an(i))

algebraically and topologically.

For every K¨othe matrix A, the spaces λp(A),1≤p≤ ∞ and p= 0, are

Fr´echet spaces.

Taking V= (vn(i))n∈Nto denote the associated decreasing sequence of func-

tions vn(i) = 1

an(i), we put

kp(V) = indilp(vn(i)),1≤p≤ ∞,

and

k0(V) = indic0(vn(i)).

These are K¨othes‘s co-echelon spaces. Define

V=λ∞(A)+={v∈(R+)N; sup

i∈N

vn(i)<∞for each n∈N}

and

Kp(V) = λp(V) = projv∈Vlp(v)

for 1 ≤p≤ ∞ as well as

K0(V) = projv∈Vc0(v).

Remark 1.69. (Bierstedt, Meise, Summers [15], 1.5) kp(V) is continuously

embedded in Kp(V).

Let E, F be locally complete locally convex spaces. Lb(E, F) denotes the

space of all continuous linear mappings from Eto F, endowed with the strong

topology. For Vand a locally convex space E, it is clear how K∞(V , E) is

defined.

Proposition 1.70. [5] Let Edenote a locally complete locally convex space.

Then there is a canonical topological isomorphism

K∞(V , E) = Lb(λ1(A), E)

of K∞(E)onto the space of all continuous linear mappings from λ1(A)into

E, endowed with the topology of uniform convergence on the bounded subsets

of λ1(A). In particular, K∞(V) = (λ1(A))0

b(see [15],2.7).

Now we come to the main example:

Example 1.71. Let B= (bk(i))k∈N, i ∈Nand C= (cn(j))n∈N, j ∈N

be K¨othe matrices. Consider the space Lb(λ1(B), λ∞(C)). We can write

λ1(B) = projkl1(bk(i)) and λ∞(C) = projnl∞(cn(j)).Suppose that λ1(B)

is distinguished, which holds if and only if (bk(i))k∈Nsatisfies condition (D)

(see [4], 6.). Bierstedt, Meise, Summers (see [15], 2.8) proved that λ1(B) dis-

tinguished is equivalent to (λ1(B))0

b∼

=k∞(W). With this result and taking

W= (wk(i))k∈N, wk(i) := bk(i)−1, we obtain that

λ1(B)0

b∼

=k∞(W) = indkl∞(wk(i)).

Now we have Lb(λ1(B), λ∞(C)) = projkLb(λ1(B), l∞(cn(j))). Since λ1(B)

is distinguished, Lb(λ1(B), l∞(cn(j))) is ultrabornological (see [5]) and by

Bierstedt, Bonet ([4], 6. and 7.)

Lb(λ1(B), l∞(cn(j))) ∼

=indkLb(l1(bk(i)), l∞(cn(j)))

∼

=indkl∞(wk(i), l∞(cn(j)))

= indkl∞(wk(i)⊗cn(j)).

holds. This implies that

Lb(λ1(B), λ∞(C)) = projnindkl∞(wk(i)⊗cn(j))

holds algebraically and topologically. If λ1(A) is distinguished, the space

Lb(λ1(B), λ∞(C)) is of the form AC(X) with X=N×Nand an,k(i, j) :=

bk(i)⊗cn(j). An easy argument shows that the isomorphism above induces a

linear isomorphism between the space LB(λ1(B), λ∞(C)) of bounded linear

maps and VC(X) = indkprojnCan,k(X).

Corollary 1.72. If A= ((an,k)n∈N)k∈Nis defined by an,k(i, j) = 1

bk(i)⊗cn(j),

i, j ∈N, and satisfies condition (Q), then the space Lb(λ1(B), λ∞(C)) is

barrelled.

Proof. For a general (P LB)-space AC(X) we have proved in theorem 1.52

that condition (Q) is satisfied if and only if Proj1A= 0. With the general

theory of (DF)-spaces (see theorem 1.34) it follows that Lb(λ1(B), λ∞(C))

is barrelled.

Corollary 1.73. If Lb(λ1(B), λ∞(C)) is barrelled, then A= ((an,k)n∈N)k∈N

defined by an,k(i, j) = 1

bk(i)⊗cn(j), i, j ∈N, satisfies condition (wQ).

Proof. With corollary 1.60 from AC(X) barrelled it follows that the space

A0C(X) is quasibarrelled (corollary 1.60). Again with the general theory

of projective spectra of (LB)-spaces (see [43]) this implies that A0C(X) is

barrelled, and condition (wQ) follows with theorem 1.48.

2 Weighted spaces of holomorphic functions

on the half-plane

2.1 Introduction to part 2

The second chapter of this work deals with weighted Banach spaces of holo-

morphic functions on the upper half-plane G. Let v:G→R+be a strictly

positive, continuous function (weight). The space Hv0(G) is defined as fol-

lows:

Hv0(G) := {f∈H(G); v|f|vanishes at infinity on G}.

This chapter is motivated by a question of Bierstedt [3]. In a survey about

weighted inductive limits of spaces of holomorphic functions he asked if the

space Hv0(G) has the approximation property under some conditions of Holt-

manns [25]. The problem remains open in general, but we give a positive

answer for weights with two additional conditions. Actually we can then

even show the existence of a basis.

In section 2.2 we give the necessary notation and an overview about results

for weighted spaces of holomorphic functions on certain domains. The main

result (theorem 2.13) is given in section 2.3. Next, in section 2.4, we present

some preparations before we give the proof of theorem 2.13 in section 2.5. In

the last section we give some examples for the weights.

An article similar to this part, except for section 2.2, is accepted for pu-

plication in Bull. Soc. R. Sci. Li`ege.

2.2 Notation and known results

Let G⊂Cor CN, N ∈N, and v:G→R+be a weight on G, i.e. a strictly

positive, continuous function. Define

Hv(G) := {f∈H(G); ||f||v:= sup

z∈G

v(z)|f(z)|<∞},

Hv0(G) := {f∈H(G); vf vanishes at infinity on G}.

Hv0(G) is a closed subspace of Hv(G), and both spaces are complete, hence

Banach spaces, where Hv0(G) carries the induced norm.

The unit balls of these spaces are denoted as follows:

B:= {f∈Hv(G); sup

z∈G

v(z)|f(z)| ≤ 1},

B0:= {f∈Hv0(G); sup

z∈G

v(z)|f(z)| ≤ 1}.

In 1993 Bierstedt, Bonet, Galbis [10] investigated weighted spaces of holo-

morphic functions for radial weights von balanced domains Gand proved

the metric approximation property of Hv0(G) if Hv0(G) contains the poly-

nomials. For starshaped domains and admissible weights Kaballo and Vogt

[27] had already proved the approximation property by a different method.

More recently Stanev [40] studied weighted spaces of holomorphic functions

on the upper half-plane. He gave a characterisation when these spaces are not

trivial. In her thesis Holtmanns [25] investigated biduals of weighted spaces

of holomorphic functions on the upper-half plane G. She introduced natural

conditions on the weight vsuch that Hv0(G)00 and Hv(G) are isometrically

isomorphic. We now present some more details of the results mentioned so

far.

Definition 2.1. A Banach space Xhas the approximation property (a.p.) if

for any compact subset M⊂Xand any ε > 0 there is a linear finite rank

operator L:X→Xwith ||Lx −x|| ≤ εfor every x∈M. If there is λ≥1

such that in addition Lcan always be chosen with ||L|| ≤ λ, then Xhas the

bounded approximation property (b.a.p.). If λcan be chosen to be 1, one says

that Xhas the metric approximation property (m.a.p.).

Definition 2.2. Let Gbe a starshaped bounded open set around zero in CN,

which means that G⊂Gρ:= {z∈CN;ρz ∈G}for 0 <ρ<1. A weight

v:G→R+with limz→∂G v(z) = 0 is called admissible if v(z)≤v(ρz) holds

for all z∈Gand 0 ≤ρ≤1.

Kaballo, Vogt [27] presented the following result in 1980:

Theorem 2.3. Let G⊂Cbe a starshaped bounded open set around zero

and v:G→R+be an admissible weight on G. Then Hv0(G)has the

approximation property.

The theorem was proved by use of the operator Tρ:Hv0(G)→H(Gρ),

(Tρf)(z) = f(ρz). The space H(Gρ) has the approximation property, and

showing that for ρ→1 the operator Tρtends to the identity uniformly on the

compact subsets of Hv0(G), it follows that Hv0(G) has the approximation

property. This proof also shows that Hv0(G) has the bounded approximation

property if A(G) := {f∈C(G); f|Gholomorph}has. An obvious example

for this situation is the unit disc in C, but A(G) also has the bounded ap-

proximation property for bounded balanced domains G⊂CN.

Definition 2.4. Let Gbe a balanced open subset of CN. A weight v:G→R

is called radial if v(λz) = v(z) for all z∈Gand all λ∈C,|λ|= 1.

Theorem 2.5. (Bierstedt, Bonet, Galbis [10]) Let Gbe a balanced open

subset of CN,v:G→R+be a radial weight and let Hv0(G)contain all the

polynomials. Then Hv0(G)has the bounded approximation property, and the

polynomials are dense in Hv0(G).

In the proof of theorem 2.5 the authors used the Ces`aro means of the partial

sums of the Taylor series about 0 to construct linear operators of finite rank

from Hv(G) into Hv0(G).

To answer the question of Bierstedt, it is not possible to use the same argu-

ments and ideas as in the case of radial weights on balanced domains.

Now let Gbe the upper half-plane, G={z∈C; Imz > 0}. Stanev [40]

presented conditions under which weighted spaces of holomorphic functions

on the upper half-plane are not trivial. His notation is different from the

usual one. He considered functions p:R+→R+with inft∈[1

c,c]p(t)>0 for

all c > 1 and the norm

||f||p:= sup

z∈G

p(Imz)|f(z)|.

Theorem 2.6. (Stanev [40]) Let p:R+→R+be a function as above and

put v(z) = vp(z) := p(Imz), z ∈G.

i) Hv(G)6={0}if and only if there exist a, b ∈Rsuch that

(−1) ln p(t)≥at +b

for all t > 0.

ii) Hv0(G)6={0}if and only if the following two conditions on the func-

tion pare satisfied:

(a) there exist a, b ∈Rsuch that (−1) ln p(t)≥at +bfor all t > 0,

(b) limt→0+p(t) = 0.

Next we present a result of Holtmanns for weighted spaces of holomorphic

functions on the upper half-plane Gand their biduals. In her proof she used

a general result of Bierstedt, Summers which we will give first.

Proposition 2.7. (Bierstedt, Summers [16]) If B0is dense in Bin the

compact open topology , then Hv(G)is isometrically isomorphic to the bidual

Hv0(G)00.

Theorem 2.8. (Holtmanns [25]) Let Gbe the upper half-plane and let vbe

a continuous weight on Gsuch that:

(i) v > 0on G,

(ii) limImz→0v(z) = 0,

(iii) there exists 0< r0<1with v(z)≤v(z+ir)for all z∈Gand

0< r ≤r0.

Then Hv0(G)00 and Hv(G)are isometrically isomorphic.

For f∈Hv0(G) Holtmanns introduced auxiliary functions

fn(z) := f(z+1

n)n

z+i, z ∈G, n ∈N

(as in the proof of the classical Phragmen-Lindel¨of theorem) to prove the

condition of proposition 2.7 in her case. The operators f→fnon Hv0(G)

will be important for the proof of our main result, too.

2.3 Commuting b.a.p. and the main result

While we tried to solve the problem of Bierstedt, it turned out that two

additional conditions on the weights were needed. With these conditions

and a result of Lusky [30] it was even possible to show the existence of a

basis.

Definition 2.9. Let Xbe a Banach space. A sequence (ej)j∈Nis called

Schauder basis of Xif for each x∈Xthere is a uniquely determined sequence

(ξj(x))j∈Nin IK, for which x=P∞

j=1 ξj(x)ejis true.

Definition 2.10. Let Xbe a Banach space. A sequence of bounded linear

operators Vn:X→Xof finite rank is called commuting approximating

sequence (c.a.s.) if limn→∞ Vnx=xfor each x∈Xand VnVm=Vmin(n,m)

whenever n6=m. If there exists such a sequence (Vn)n∈Non X, then X

is said to have the commuting bounded approximation property (CBAP). If

VnVm=Vmin(n,m)holds, in addition, even for n=mthen Xis said to have

afinite dimensional Schauder decomposition (FDD).

Clearly, by the Banach-Steinhaus theorem (CBAP) implies the bounded ap-

proximation property. It is known that there are Banach spaces with (CBAP)

which do not have (FDD).

Definition 2.11. Let Xbe a given Banach space. For a fixed pwith 1 ≤

p≤ ∞ we say that a sequence of continuous linear operators Vn:X→X

factors uniformly through lm

p’s with respect to λif there are suitable integers

mn∈Nand continuous linear operators

Tn:X→lmn

p, Sn:lmn

p→X,

with

Vn=SnTn,sup

||Tn|| ≤ λand sup

||Sn|| ≤ λ.

In 1996 Lusky [30] presented the following result which we will use in the

case p=∞to show that Hv0(G) has a basis.

Theorem 2.12. Let the Banach space Xhave a commuting approximating

sequence (Vn)n∈Nsuch that Vn−Vn−1factors uniformly through lm

p’s for some

1≤p≤ ∞. Then Xhas a basis.

From now on Gis the upper half-plane. For our main result we need the

following conditions on the weight v. Let v:G→Rbe continuous such that

(i) v > 0 on G,

(ii) limImz→0v(z) = 0,

(iii) there exists 0 < r0<1 with v(z)≤v(z+ir) for all z∈Gand

0< r ≤r0,

(iv) for each ε > 0 there exists b=b(ε)>0 such that v(z)≥bfor all z∈G

with Imz≥ε,

(v) vis bounded.

The first three conditions were introduced by Holtmanns [25]. She did not

require conditions (iv) and (v) for her work, but these conditions seem to be

necessary for our result. The following is the main result of the second part

of this work.

Theorem 2.13. Let Gbe the upper half-plane and va weight on Gwhich

satisfies conditions (i)-(v) above. Then Hv0(G)has a basis.

With theorem 2.12 above the proof of theorem 2.13 is reduced to showing

that Hv0(G) has a commuting approximating sequence {Vn}∞

n=1 such that

Vn−Vn−1factors uniformly through lm

∞’s.

2.4 Preparations

In the sequel some technical tools are given which are needed for the proof.

In her thesis [25] Holtmanns defined linear operators Θnas follows:

Definition 2.14. For f∈Hv0(G) let

Θn:Hv0(G)→Hv0(G), n ∈N,Θnf:= fn

with fn(z) := f(z+i

n)n

z+ifor z∈G.

The main branch of the n-th root is well-defined since z→1

z+imaps Ginto

the set {z∈C; Imz < 0 and |z|<1}. The functions fnare holomorphic on

Gsince z+i6= 0 for all z∈G.

Lemma 2.15. (Holtmanns [25]) Θnis well-defined and continuous as an

operator from Hv0(G)into Hv0(G).Θnfconverges to fin the compact-

open topology, f∈Hv0(G), since |n

z+i| → 1for n→ ∞.

Lemma 2.16. Let f∈Hv0(G)and Θnbe as defined before. For each ε > 0

there exist n0∈Nand a compact set K⊂Gwith v(z)|Θnf(z)−f(z)| ≤ ε

for all z∈G\Kand for any fixed n∈N, n ≥n0.

Proof. Let ε > 0 be given. Set ˜ε=1

4ε.f∈Hv0(G) means that there exist

L > 0 and 0 < l < 1

2with

v(z)|f(z)| ≤ ˜ε∀z∈G\[−L, L]×i[l, L].

Set K:= [−L, L]×i[l

2, L]. For all z∈G\Kthe following inequality holds

for n∈Nlarge enough such that condition (iii) can be applied:

v(z)|Θnf(z)−f(z)| ≤ v(z)(|fn(z)−f(z+i

n)|+|f(z+i

n)−f(z)|)

≤v(z)|f(z+i

n)n

z+i−f(z+i

n)|

+v(z)|f(z+i

n)|+v(z)|f(z)|

≤v(z+i

n)|f(z+i

n)|| n

z+i−1|

+v(z+i

n)|f(z+i

n)|+v(z)|f(z)|.

Let us now show that v(z+i

n)|f(z+i

n)| ≤ ˜εfor n∈Nlarge enough. Two

cases are possible:

Case 1: |Rez|> L or Imz > L. Then z /∈K⇒z+i

n/∈K⇒v(z+i

n)|f(z+

n)| ≤ ˜ε.

Case 2: Imz < l

2and |Rez| ≤ L. Then there exists n0∈Nwith 1

n<l

2for

all n∈N, n ≥n0.z+i

n=x+i(y+1

n) with y+1

n<l

2+1

n≤l

2+l

2=l

⇒Im(z+i

n)< l ⇒v(z+i

n)|f(z+i

n)| ≤ ˜ε.

On the other hand, supz∈G|n

z+i|= supz∈Gn

|z+i|= 1 ∀n∈Nsince

|z+i| ≥ |Imz|+ 1 ≥1∀z∈G, and hence |1−n

z+i| ≤ 2.

Using these two estimates in the right hand side of the above inequality

yields

v(z)|Θnf(z)−f(z)| ≤ 2˜ε+ ˜ε+ ˜ε≤ε

for each z∈G\K.

Corollary 2.17. With lemma 2.15 and lemma 2.16 it follows that for f∈

Hv0(G)and for each ε > 0there exists n0∈Nsuch that ||Θnf−f||v≤ε

for any fixed n∈N, n ≥n0.

Next we define the space A0(G), extend the operator Θnto Gand show that

fnmaps Hv0(G) to A0(G) and that there exists a restriction mapping back

to Hv0(G).

Definition 2.18. Define

A0(G) := {f∈C(G); f|G∈H(G),∀η > 0∃N∈R+:

|f(z)|< η ∀z∈G, |z| ≥ N},

endowed with the sup-norm and extend Θnfcontinuously to Gby taking

(Θnf)(x) = f(x+1

n)n

x+ifor x∈R.

Lemma 2.19. For each f∈Hv0(G)and each n∈Nwe have Θnf∈A0(G),

i.e. there exists a linear mapping

Rn:Hv0(G)→A0(G), Rnf=fn∀n∈N.

Proof. Let f∈Hv0(G) and n∈Nbe fixed. Set ε=1

n. With condition (iv)

for the weight vthere exists b=b(1

n)>0 with v(z)≥bfor all z∈Gwith

Imz≥ε. Then for each z∈G, also v(z+i

n)≥bholds. Now fix η > 0.

f∈Hv0(G) means that for ˜η:= η·bthere exists N > 0 such that

|f(z+i

n)|v(z+i

n)≤˜η

for all z∈Gwith |z| ≥ N. Then for fnand such a z∈Gthe following

estimate holds:

|fn(z)|=|f(z+i

n)|| n

z+i|

=|f(z+i

n)|v(z+i

n)1

v(z+i

n)|n

z+i|

≤˜η·1

b=η,

hence fn∈A0(G).

Lemma 2.20. The restriction mapping

R:A0(G)→Hv0(G), f →f|G,

is well-defined and continuous.

Proof. Fix f∈A0(G). By condition (v), vis bounded, i.e. there exists

M > 0 with v(z)≤Mfor all z∈G. Let η > 0 be arbitrary, but fixed. Set

η0:= η

M. For η0there exists N > 0 such that |f(z)|< η0for all z∈Gwith

|z| ≥ N. Then v(z)|f(z)|< M η

M=ηfor all z∈Gwith |z| ≥ N. Define

L:= N+ 1. By condition (ii) we can extend vcontinuously to Gby putting

˜v(z) := v(z) for z∈Gand ˜v(z) := 0 elsewhere. ˜vis uniformly continuous

on K:= [−L, L]×i[δ, L] for each δ > 0. fis bounded on Kwhich means

that there exists S > 0 such that |f(z)| ≤ Sfor all z∈K. For ε:= η

S>0

there exists δ > 0 : z, z0∈K, |z−z0|< δ ⇒ |˜v(z)−˜v(z0)|< ε. We would

like to show that v(z)|f(z)|< η for all z /∈K. The desired inequality holds

if |z| ≥ N+ 1. Let z=x+iy /∈Kand consider 0 < y < δ and |z| ≤ N+ 1.

We get |x−z|=|x−x−iy|=|y|< δ and ˜v(z) = ˜v(z)−˜v(x)< ε =η

hence v(z)|f(z)|<η

SS=ηfor all z /∈K.

Lemma 2.21. The sequence (Rn)n∈Nof linear mappings Rn:Hv0(G)→

A0(G)is uniformly bounded.

Proof. For n≥n0large enough so that condition (iii) can be applied, we get

||Rnf||v=||fn||v= sup

z∈G

|fn(z)|v(z) = sup

z∈G

|f(z+i

n)n

z+i|v(z)

≤sup

z∈G

|f(z+i

n)|v(z+i

n)|n

z+i|

≤ ||f||v.

In the next step we define the disc algebra A(D), the space A0(D), repeat

some properties of these spaces and show the existence of an isometric iso-

morphism between A0(D) and A0(G).

Definition 2.22. Let Dbe the open unit disc, D:= {z∈C;|z|<1}. Define

the disc algebra

A(D) := {f∈C(D); f|Dis holomorphic},

and the space

A0(D) := {f∈A(D); f(1) = 0}.

Because the polynomials are dense in the disc algebra one can write A0(D)

A0(D) = span{zj−1; j= 1,2, ...}.

Bockarev [17] showed in 1974:

Proposition 2.23. The disc algebra A(D)has a Schauder basis and therefore

the bounded approximation property.

Proposition 2.24. A0(D)has the bounded approximation property.

Proof. By proposition 2.23, A(D) has the bounded approximation property.

p:A(D)→A0(D), p(f) = f−f(1), f ∈A(D), is a bounded projection

onto A0(D). Because of this, A0(D) is complemented in the disc algebra and

inherits the bounded approximation property from A(D).

Proposition 2.25. There exists an isometric isomorphism Tbetween A0(G)

and A0(D).

Proof. Compare [39], p. 81. Define α:G→D, α(z) := z−i

z+ifor z∈G.αis a

linear fractional transformation of the upper half-plane Gonto the unit disc

D. The inverse mapping of αis β:D→G, β(w) := i1+w

1−w, w ∈D. For each

c≥0, αmaps the half plane Imz > c onto the disc {w;|w−c

1+c|<1

1+c},

and αmaps the line Imz=conto the circle {w;|w−c

1+c|=1

1+c}with the

point 1 deleted, also β(1) = ∞and α(∞) = 1. Now we can define

T:A0(G)→A0(D) as Tf := f◦α, f ∈A0(G),

which is an isometric isomorphism from A0(G) onto A0(D).

From now on we are following a method of Lusky (see [29]) to construct a suit-

able commuting approximating sequence (Vn)n∈N, Vn:Hv0(G)→Hv0(G)

such that Vn−Vn−1factors uniformly through lm

∞’s.

Definition 2.26. Let H(D) := {f:D→C;fcontinuous, f|Dharmonic}

endowed with the sup-norm and let f∈ H(D) have the Fourier series

f(reiϕ) = P∞

k=−∞ αkr|k|eikϕ.

Define ˜

Vn:H(D)→ H(D) as

(˜

Vnf)(reiϕ) := X

|k|≤2n

αkr|k|eikϕ +X

2n<|k|≤2n+1

2n+1 − |k|

2nαkr|k|eikϕ,

Vnis the convolution with the de la Vall´ee Poussin kernel which is defined as

Vn(z) := 2F2n+1 (z)− F2n(z),

where Fn(z) is the Fej´er kernel

F2n(z) :=

k=−n

(1 −|k|

2n)eikϕ.

and Vn:A0(D)→A0(D) as

Vnf:= ˜

Vnf−(˜

Vnf)(1) ·z2n, f ∈A0(D).

Lemma 2.27. For the Fourier series f=Pαkr|k|eikϕ we define the Ces`aro

means σn:H(D)→ H(D)by σn(f) := P|k|≤2n

2n−|k|

2nαkr|k|eikϕ, cf. [24].

Then

2σn+1(f)−σn(f) = ˜

Vn(f)

holds for each n∈N.

Proof. By calculating we obtain

2σn+1(f)−σn(f)

= 2 X

|k|≤2n+1

2n+1 − |k|

2n+1 αkr|k|eikϕ −X

|k|≤2n

2n− |k|

2nαkr|k|eikϕ

2n<|k|≤2n+1

2n+1 − |k|

2nαkr|k|eikϕ +X

|k|≤2n2n+1 − |k|

2n−2n− |k|

2nαkr|k|eikϕ

2n<|k|≤2n+1

2n+1 − |k|

2nαkr|k|eikϕ +X

|k|≤2n2n+1 −2n

2nαkr|k|eikϕ

2n<|k|≤2n+1

2n+1 − |k|

2nαkr|k|eikϕ +X

|k|≤2n

αkr|k|eikϕ

=˜

Vn(f).

Lemma 2.28. For f∈A0(D)and Vndefined as before, the following holds:

(i) limn→∞ Vnf=ffor each f∈A0(D),

(ii) dim VnA0(D)<∞,

(iii) VnVm=Vmin(n,m), if n6=m.

Proof. (i) and (ii) follow immediately from the definition of Vn, respectively

of ˜

Vnand lemma 2.27 because the Ces`aro means are convergent to fin

A(D). To show (iii), we first prove ˜

Vn˜

Vm=˜

Vmin(n,m), for n6=m. For m > n,

Vn˜

Vm=˜

Vnfollows directly from the definition. ˜

Vnzk= 0 if k≥2n+1, and

Vmzk=˜

Vnzk=zkif k≤2n<2m.

PSfrag replacements

2−m2−n2n2m

Vm, m > n

For n > m one can use the same arguments to get ˜

Vn˜

Vm=˜

Vm. To show

the desired equation for VnVm, set Wn(f) = −(˜

Vnf)(1)z2n. For m > n we

obtain:

VnVm(f) = ( ˜

Vn+Wn)( ˜

Vm+Wm)(f)

= ( ˜

Vn˜

Vm+˜

VnWm+Wn˜

Vm+WnWm)(f)

=˜

Vn(f)−˜

Vn(( ˜

Vmf)(1)z2m)−˜

Vn(˜

Vmf)(1)z2n−Wn(( ˜

Vmf)(1)z2m)

=˜

Vn(f)−(˜

Vmf)(1) ˜

Vn(z2m)−(˜

Vnf)(1)z2n

+( ˜

Vmf)(1)( ˜

Vn(z2m))(1)z2n

=˜

Vn(f) + Wn(f)

=Vn(f).

In the case m < n one uses the same arguments and obtains VnVm=Vm.

Lemma 2.29. For trigonometric polynomials Pkαkr|k|eikϕ define

P(X

αkr|k|eikϕ) := X

k≥0

αkr|k|eikϕ,

with generally unbounded P. Then

P(˜

Vn−˜

Vn−1)(f) = ei2nϕσn(e−i2nϕf)−1

2ei2n−1ϕσn−1(e−i2n−1ϕf).

Hence P(˜

Vn−˜

Vn−1)is a continuous linear operator and the same then holds

for P(Vn−Vn−1).

Proof. By some calculations we get

P(˜

Vn−˜

Vn−1)(f)

k=0

αkrkeikϕ −

2n−1

k=0

αkrkeikϕ

−

k=2n−1+1

2n−k

2n−1αkrkeikϕ +

2n+1

k=2n+1

2n+1 −k

2nαkrkeikϕ

k=2n−1+1

αkrkeikϕ −

k=2n−1+1

2n−k

2n−1αkrkeikϕ +

2n+1

k=2n+1

2n+1 −k

2nαkrkeikϕ

k=2n−1+1 1−2n−k

2n−1αkrkeikϕ +

2n+1

k=2n+1

2n+1 −k

2nαkrkeikϕ

k=2n−1+1

k−2n−1

2n−1αkrkeikϕ +

2n+1

k=2n+1

2n+1 −k

2nαkrkeikϕ

and

ei2nϕσn(e−i2nϕf)−1

2ei2n−1ϕσn−1(e−i2n−1ϕ)

|k−2n|≤2n

2n− |k−2n|

2nαkrkeikϕ −1

|k−2n−1|≤2n−1

2n−1− |k−2n−1|

2n−1αkrkeikϕ

0≤k≤2n+1

2n− |k−2n|

2nαkrkeikϕ −1

0≤k≤2n

2n−1− |k−2n−1|

2n−1αkrkeikϕ

k=0

2n−2n+k

2nαkrkeikϕ +

2n+1

k=2n+1

2n−k+ 2n

2nαkrkeikϕ

−

2n−1

k=0

2n−1−2n−1+k

2nαkrkeikϕ −

k=2n−1+1

2n−1−k+ 2n−1

2nαkrkeikϕ

k=0

2nαkrkeikϕ +

2n+1

k=2n+1

2n+1 −k

2nαkrkeikϕ

−

2n−1

k=0

2nαkrkeikϕ −

k=2n−1+1

2n−k

2nαkrkeikϕ

k=2n−1+1 k

2n−2n−k

2nαkrkeikϕ +

2n+1

k=2n+1

2n+1 −k

2nαkrkeikϕ

k=2n−1+1

k−2n+k

2nαkrkeikϕ +

2n+1

k=2n+1

2n+1 −k

2nαkrkeikϕ

k=2n−1+1

k−2n−1

2n−1αkrkeikϕ +

2n+1

k=2n+1

2n+1 −k

2nαkrkeikϕ.

Proposition 2.30. Vn−Vn−1factors uniformly through lm

∞’s on A0(D).

Proof. By the definition of the Ces`aro means, ||σn|| = 1 holds for all n∈N;

again cf. [24]. With lemma 2.27 we obtain || ˜

Vn|| ≤ 3 for all n∈N. Hence

(Vn)nis uniformly bounded. C(∂D) is a L∞-space, and it is well-known that

H(D) is isometrically isomorphic to C(∂D). Hence H(D) is a L∞-space.

There exists λ > 0 such that for each n∈Nthere is F⊂ H(D) with

Vn+1H(D)⊂Fand there is an isomorphism Φ : F→lM

∞with M= dim F <

∞and ||Φ||·||Φ−1|| ≤ λ. Note that A0(D)⊂ H(D). Define Tn:A0(D)→lM

∞

Tnf:= Φ(Vn+1 −Vn−2)f,

and Sn:lM

∞→A0(D) by

Sng:= P(Vn−Vn−1)Φ−1g−(P(Vn−Vn−1)Φ−1g)(1).

We have supn||Sn|| <∞,supn||Tn|| <∞and

SnTn(f) = SnΦ(Vn+1 −Vn−2)f

=P(Vn−Vn−1)(Vn+1 −Vn−2)f

−(P(Vn−Vn−1)(Vn+1 −Vn−2)f)(1)

=P(Vn−Vn−1)f−(P(Vn−Vn−1)f)(1) = (Vn−Vn−1)f

where the last but one equality holds because of

(Vn−Vn−1)(Vn+1 −Vn−2) = VnVn+1 −VnVn−2−Vn−1Vn+1 +Vn−1Vn−2

=Vn−Vn−2−Vn−1+Vn−2=Vn−Vn−1.

2.5 Proof of theorem 2.13

Before we give the proof of theorm 2.13 we will collect the results of section

2.4 in the following diagram. We have

Hv0(G)Rn

−→ A0(G)T

−→ A0(D)Vn

−→ A0(D)T−1

−→ A0(G)R

−→ Hv0(G).

With the linear mapping

Rn:Hv0(G)→A0(G), Rnf=fn∀n∈N,

the isometric isomorphism

T:A0(G)→A0(D), Tf := f◦β, f ∈A0(G),

the commuting approximating sequence (Vn)n

Vn:A0(D)→A0(D), Vnf:= ˜

Vnf−(˜

Vnf)(1) ·z2n, f ∈A0(D)

and the restriction mapping

R:A0(G)→Hv0(G), f →f|G.

Now we come to the proof of theorem 2.13: For a suitable sequence (mn)n∈N

of indices we can assume without loss of generality:

(∗)RmnRT −1(r|k|eikϕ −1) = T−1(r|k|eikϕ −1) ∀ |k| ≤ 2n+1.

If (∗) is not true, replace Rmnby

Rmn:= Rmn(id −Pn) + R−1Pn

= (R−1−Rmn)Pn+Rmn,

with En:= span{RT −1(r|k|eikϕ −1); |k| ≤ 2n+1},En⊂Hv0(G) and Pn:

Hv0(G)→Ena bounded projection. Then

RmnRT −1= (R−1−Rmn)RT−1+RmnRT−1=T−1

holds on En, but we have to show that ˜

Rmnis uniformly bounded. By

corollary 2.17, one can choose m1< m2< ... with

||RmnRT −1(r|k|eikϕ −1) −T−1(r|k|eikϕ −1)|| ≤ 1

n2n+2||Pn||w

for all |k| ≤ 2n+1, where w:= ||R−1|En||. By the definition of ˜

Rmnwe obtain

|| ˜

Rmn−Rmn|| =||(R−1−Rmn)Pn||.

Let x∈Enwith ||x||v= 1. One can write xas

x:= X

|k|≤2n+1

αkRT −1(r|k|eikϕ −1).

With U:= (R−1−Rmn)Pnone gets

||Ux||v≤X

|k|≤2n+1

|αk| · ||URT−1(r|k|eikϕ −1)||v.

Define Fn:= span{(r|k|eikϕ −1); |k| ≤ 2n+1}. Then Fn⊂A0(D), RT−1Fn=

Enand ||(RT −1|Fn)−1|| ≤ w||T|| holds. Set W:= (RT −1|Fn)−1and note and

Wx =X

|k|≤2n+1

|αk|(r|k|eikϕ −1).

Here the Fourier coefficients can be estimated as follows:

|αk| ≤ ||W x|| ≤ ||W|| · ||x||v=||W|| ≤ w||T||.

Putting the estimates together we obtain

|| ˜

Rmn−Rmn|| = sup{||Ux||v;||x||v= 1}

≤X

|k|≤2n+1

|αk| · ||URT −1(r|k|eikϕ −1)||v

≤2n+2w||T|| · ||T−1(r|k|eikϕ −1) −RmnRT−1(r|k|eikϕ −1)||

≤||T||

n||Pn||.

Now define ˆ

Vn:Hv0(G)→Hv0(G) by

Vn:= RT −1VnT Rmn.

Comparing the definition of ˆ

Vnwith the diagram at the beginning of this

section we obtain that ˆ

Vnis welldefined. We claim that ˆ

Vnis a commuting

approximating sequence with ˆ

Vnˆ

Vm=ˆ

Vmin(n,m)for n6=m, dim ˆ

VnHv0(G)<

∞and limn→∞ ˆ

Vnf=ffor f∈Hv0(G). Let n > m; then we have:

Vnˆ

Vm=RT −1VnT RmnRT−1VmTRmm

=RT −1VnT T −1VmT Rmm

=RT −1VnVmT Rmm

=RT −1VmT Rmm

=ˆ

Vm.

This holds because of (∗) and because T T −1is the identity on A0(D). If

n < m we obtain ˆ

Vnˆ

Vm=ˆ

Vnby the same arguments. In proposition 2.30 we

showed that there exist kn, Tn:A0(D)→lkn

∞and Sn:lkn

∞→A0(D) with

supn||Sn|| <∞,supn||Tn|| <∞and SnTn=Vn−Vn−1. Set

Tn:Hv0(G)→lkn

∞,ˆ

Tn:= TnTRmn,

Sn:lkn

∞→Hv0(G),ˆ

Sn:= RT −1Sn.

With (∗) and the definition of Vnit follows that

(∗∗)VnTRmj=VnTRmn

holds for all j≥nsince VnTRmnRT −1(r|k|eikϕ −1) = VnT T −1(r|k|eikϕ −1) =

Vn(r|k|eikϕ −1) for each |k| ≤ 2n+1. Note that supn|| ˆ

Sn|| <∞,supn|| ˆ

Tn|| <

∞and by (∗∗)

Snˆ

Tn=ˆ

Sn(TnTRmn)

=RT −1SnTnT Rmn

=RT −1(Vn−Vn−1)T Rmn

= (RT −1Vn−RT −1Vn−1)T Rmn

=RT −1VnT Rmn−RT −1Vn−1T Rmn−1

=ˆ

Vn−ˆ

Vn−1.

We have constructed a commuting approximating sequence ˆ

Vnsuch that

Vn−ˆ

Vn−1factors uniformly through lm

∞’s. With theorem 2.12 it follows that

Hv0(G) has a basis.

2.6 Examples

Example 2.31. (Stanev [40])

i) Let p:R+→R+be defined by p(t) = 1 for all t∈R+and define

v(z) = vp(t) := p(Imz) for z∈G. In this case Hv(G) = H∞(G) and

Hv0(G) = {0}, because condition ii) of theorem 2.6 is not satisfied.

ii) Let p:R+→R+be defined by p(t) = exp(t2) for all t∈Rand define

v(z) = vp(t) := p(Imz) for z∈G. From theorem 2.6 it follows that

Hv(G) = {0}and Hv0(G) = {0}.

Example 2.32. Let Gbe the upper half-plane and v:G→Rbe defined

by v(z) := (Imz)rfor Imz≤1 and v(z) := 1 elsewhere, r > 0. vsatisfies the

conditions (i) - (v). Hence Hv0(G) has a basis.

Example 2.33. Let Gbe the upper half-plane and v:G→Rbe defined by

v(z) := exp(−1/(Imz)2).It is easy to see that vsatisfies conditions (i)-(v).

Hence Hv0(G) has a basis.

Example 2.34. Let Gbe the upper half-plane and v:G→Rbe defined

by v(z) := Imz.vsatisfies conditions (i)-(iv), but vis not bounded. But

Hv0(G) has the bounded approximation property.

Proof: The idea of this construction goes back to Stanev [40]. Let the

weight won the unit disc Dbe defined by w(δ) := (1 − |δ|2). wis ra-

dial and lim|δ|→1w(δ) = 0. Hence Hw0(D) has the bounded approximation

property [10]. For f∈Hw0(D) we define the operator ˜

T:Hw0(D)→

Hv0(G),˜

Tf(z) = (f◦˜

β)(z)·(4

(1−iz)2), z ∈Gwith ˜

β(z) = 1+iz

1−iz for z∈G.

βmaps the upper half-plane Gonto the unit disc D. The operator ˜

Tis a

topological isomorphism from Hw0(D) onto Hv0(G) [40].

For z=x+iy ∈Gset ˜

β(z) = 1+iz

1−iz =δand calculate 1 − |δ|2:

1− |δ|2= 1 − |1+iz

1−iz |2=|1−iz|2−|1+iz|2

|1−iz|2=(1−iz)(1+z)−(1+iz)(1−z)

|1−iz|2

=1+iz−iz+|z|2−(1−iz+iz+|z|2)

|1−iz|2=2iz−2iz

|1−iz|2

=2ix+2y−2ix+2y

|1−iz|2=4y

|1−iz|2

=4Imz

|1−iz|2

For f∈Hw0(D) the following holds:

f∈Hw(D)⇔(1 − |δ|2)p|f(δ)|<∞ ∀δ∈D

⇔(4Imz

|1−iz|2)p|f(1+iz

1−iz )|<∞ ∀z∈G

⇔v(z)|˜

Tf(z)|<∞∀ z∈G

⇔˜

Tf ∈Hv(G) and

f∈Hw0(D)⇔f∈Hw(D),lim|δ|→1−(1 − |δ|2)p|f(δ)|= 0

⇔˜

Tf ∈Hv(G),limImz→0(4Imz

(1−iz)2)pf(1+iz

1−iz ) = 0

⇔˜

Tf ∈Hv(G),limImz→0v(z)Tf(z) = 0

⇔˜

Tf ∈Hv0(G).

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