Spaces of continuous and holomorphic functions with growth conditions / Simone Agethen [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.

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