Fi s Ad anced Cou se in Ope a o Theo y and Complex Analysis,
Uni e si y o Se ille, June 2004
ON SPECTRAL STRUCTURE OF BOUNDED LINEAR
OPERATORS ON REFLEXIVE BANACH SPACES
STANISLAV A. SHKARIN AND OLEG G. SMOLYANOV
Abs ac . A desc ip i e cha ac e iza ion o poin , con inuous, and esid-
ual spec a o ope a o s ac ing on a sepa able Hilbe space is ob ained.
The possible poin spec a o bounded linea ope a o s ac ing on `p,
1<p<∞a e cha ac e ized.
1. In oduc ion
As usual Cis he ield o complex numbe s, Ris he ield o eal numbe s,
Zis he se o in ege s and Nis he se o posi i e in ege s. We also deno e
N=N∪ {∞}. All ec o spaces in his pape a e assumed o be o e he ield
Cand all opological ec o spaces a e assumed o be Hausdo . Fo a ec o
space Xand a linea ope a o T:DT−→ Xde ined on a linea subspace DT
o X, he se
σp(T) = {z∈C: dim {x∈DT:Tx =zx}=ν(z)>0}
is called he poin spec um [2] o T. The numbe ν(z)∈N o z∈σp(T) is
called he mul iplici y o z. Fo any n∈Nwe deno e
σp,n(T) = {z∈σp(T) : ν(z) = n}and σn
p(T) = {z∈σp(T) : ν(z)⩾n}.
No e ha σp(T) = σ1
p(T) and σ∞
p(T) = σp,∞(T).
Fo a opological ec o space X, an ope a o T:DT−→ Xis said o be
closed [2] i i s g aph ΓT={(x, Tx) : x∈DT}is closed in X×Xand Tis called
2000 Ma hema ics Subjec Classi ica ion. 47A10.
Re ised Sep embe 14, 2004.
Pa ially suppo ed by “El Minis e io de Ciencia y Tecnolog´ıa, Spain” (BFM2003-00034)
and by “Jun a de Andaluc´ıa” (FQM-260).
133
134 S. A. SHKARIN and O. G. SMOLYANOV
densely de ined [2], i DTis dense in X. The spec um o T:DT⊆X−→ X
is [2] he se
σ(T) = C ½z∈C: he ope a o T−zI has con inuous
densely de ined in e se ¾,
whe e I:X−→ Xis he iden i y ope a o . The se s
σc(T) = {z∈σ(T) σp(T) : he se (T−zI)(DT) is dense in X}and
σ (T) = {z∈σ(T) σp(T) : he se (T−zI)(DT) is no dense in X}
a e called con inuous spec um and esidual spec um [2] o T, espec i ely.
Ob iously, σ(T) is he disjoin union o h ee se s σp(T), σc(T) and σ (T).
Fo any non-emp y compac se K⊂R, Kalisch [3] cons uc ed a bounded
linea ope a o Tac ing on a sepa able Hilbe space such ha σ(T) = σp(T) =
K. Using a simila cons uc ion, Nikolskaia [8] p o ed ha a se A⊂Cis he
poin spec um o a linea con inuous ope a o s on a sepa able Hilbe space i
and only i Ais a bounded Fσ-se . R. Kau mann [4, 5, 6, 7] p o ed ha a se
A⊂Cis a poin spec um o a bounded linea ope a o on a sepa able Banach
space i and only i Ais bounded and is a Souslin se , ha is, he con inuous
image o a comple e sepa able me ic space. The esul o Kau mann was
s eng hened by he au ho s in he ollowing way [11].
Theo em S. Le A, B, C be h ee disjoin subse s o C. Then he ollowing
condi ions a e equi alen .
(S1) The e exis s a bounded linea ope a o Tac ing on a sepa able Banach
space Xsuch ha A=σp(T),B=σc(T)and C=σ (T).
(S2) The se A∪B∪Cis non-emp y and compac , he se s A,C Band
C Ca e Souslin and he e exis s an Fσ-se Dand an ope a o Tsuch
ha σp(T)∪D=σp(T)∪σ (T).
In he p esen wo k we p o ide he ollowing cha ac e iza ion o he se s
σp,n(T), σn
p(T), σc(T) and σ (T) o bounded and o closed densely de ined
linea ope a o s ac ing on a sepa able Hilbe space.
Theo em 1. I. Le Tbe a closed densely de ined linea ope a o ac ing on a
sepa able Hilbe space H. Then σn
p(T)is an Fσ-se o any n∈Nand σc(T)
is a Gδ-se .
II. Le K⊂Cbe non-emp y and compac and K=A∪B∪C, whe e
A∩B=A∩C=B∩C=∅,Ais an Fσ-se and Bis a Gδ-se . Le also
Anbe a dec easing sequence o Fσ-se s such ha A=A1. Then he e exis s
a bounded linea ope a o Tac ing on a sepa able in ini e dimensional Hilbe
space such ha σn
p(T) = An o any n∈N,σc(T) = Band σ (T) = C.
III. Le K⊂Cbe a closed se and K=A∪B∪C, whe e A∩B=A∩C=
B∩C=∅,Ais an Fσ-se and Bis a Gδ-se . Le also Anbe a dec easing
sequence o Fσ-se s such ha A=A1. Then he e exis s a closed densely
de ined linea ope a o Tac ing on a sepa able in ini e dimensional Hilbe
space such ha σn
p(T) = An o any n∈N,σc(T) = Band σ (T) = C.
BOUNDED LINEAR OPERATORS ON REFLEXIVE BANACH SPACES 135
Since he spec um o a bounded linea ope a o on a Hilbe space is non-
emp y and compac and he spec um o a closed densely de ined linea ope a o
on a Hilbe space is closed, Theo em 1 p o ides a comple e desc ip ion o all
possible σn
p(T), σc(T) and σ (T) o bounded and o closed densely de ined
ope a o s ac ing on `2. No e also ha Theo em 1, e en in i s poin spec um
pa , can no be ob ained om he cons uc ions used by Nikolskaia o Kalisch,
because he la e do no p o ide he ull a ie y o he pa s σn
p(T) o he
poin spec um. Un o una ely he p oo o Theo em 1 does no admi any
s aigh o wa d modi ica ion applicable o any single Banach space di e en
om `2. The ques ion whe he Theo em 1 emains ue i one eplaces he
sepa able Hilbe space by, o ins ance, `p o 1 <p<∞,p6= 2 emains
open. Howe e , using a comple ely di e en app oach, we p o e an analogue
o Nikolskaia’s heo em o hese spaces.
Theo em 2. Le 1< p < ∞and A⊂C. Then he e exis s a bounded
linea ope a o T:`p−→ `p o which σp(T) = Ai and only i Ais a bounded
Fσ-se .
No e also ha he e a e sepa able e lexi e Banach spaces X, o which
he amily o he se s σp(T), σc(T) and σ (T) o bounded linea ope a o s T
ac ing on Xis much poo e han o X=`2. Fo ins ance, i one akes X
being he edi a ily indecomposable [1], hen σ (T)∪σc(T)⊆ {0}and σp(T) is
coun able o any bounded linea ope a o Tac ing on X.
I is also wo h no ing ha he se s appea ing as spec a o con inuous linea
ope a o s ac ing on a sepa able F ´eche space we e cha ac e ized by Shka in
[10] (necessa y condi ions on a se o be such a spec um we e ea lie ob ained
by Slodowski [12]); namely, A⊂Cis he spec um o some linea con inuous
ope a o ac ing on a sepa able F ´eche space i and only i Ais a Gδσ-se .
2. P ope ies o spec al pa s o gene al
sepa able e lexi e Banach spaces
P oposi ion 1. Le Xbe a opological ec o space and T:DT−→ Xbe a
linea ope a o whose g aph is a union o coun ably many me izable compac
se s. Then o any n∈N,σn
p(T)is an Fσ-se .
P oo . Le n∈Nand A,Bbe he se s de ined by he o mulas
A={((x1, y1),...,(xn, yn), z)∈Γn
T×C:yj=zxj o 1 ⩽j⩽n},
B=½((x1, y1),...,(xn, yn), z)∈A: he ec o s x1, . . . , xn
a e linea ly independen ¾.
Since Ais closed in he space Γn
T×C, which is a union o coun ably many
me izable compac se s, he se Ais i sel a coun able union o me izable
compac se s. One can easily e i y ha Bis open in A. Since an open subse
o a me izable compac se is a coun able union o me izable compac se s,
136 S. A. SHKARIN and O. G. SMOLYANOV
we ha e ha Bis a coun able union o me izable compac se s. Le now
ϕ:B−→ C, ϕ((x1, y1), . . . , (xn, yn), z) = z.
Since ϕis con inuous and a con inuous image o a compac se is again a
compac se , we ha e ha he se ϕ(B) = σn
p(T) is σ-compac and, he e o e,
is an Fσ-se . ¤
Co olla y 1. Le a opological ec o space Xbe a coun able union o me iz-
able compac se s and T:DT−→ Xbe a closed linea ope a o . Then o any
n∈N,σn
p(T)is an Fσ-se .
Fo a locally con ex opological ec o space X, he symbol X0s ands o
he space o linea con inuous unc ionals on X. As usual o y∈X0and x∈X
we w i e (x, y) ins ead o y(x). I T:DT−→ Xis a densely de ined linea
ope a o , hen he symbol T0s ands o he dual ope a o T0:DT0−→ X0,
ha is, DT0is he se o ϕ∈X0 o which he unc ional x7→ (Tx, ϕ) is
con inuous on DTwi h espec o he opology o Xand T0ϕ∈X0is he
(unique) con inuous ex ension o his unc ional. No e ha he ope a o T0is
always closed when X0is endowed wi h he ∗-weak opology σ(X0, X) (see, o
ins ance, [9]).
Co olla y 2. Le Xbe a sepa able me izable locally con ex opological
ec o space and T:DT⊆X−→ Xbe a densely de ined linea ope a o .
Then o any n∈N,σn
p(T0)is an Fσ-se .
P oo . Le {Un:n∈N}be a base o neighbo hoods o ze o in X. Then X0is
he union o he se s U◦
n={y∈X0:|(x, y)|⩽1 o any x∈Un}. Alaoglu’s
heo em [9] implies ha U◦
na e compac in he ∗-weak opology σ(X0, X). Since
Xis sepa able, he compac spaces (U◦
n, σ(X0, X)) a e me izable [9]. Since T0
is a closed ope a o on (X0, σ(X0, X)), i emains o apply Co olla y 1. ¤
P oposi ion 2. Le Xbe a locally con ex opological ec o space and
T:DT−→ Xbe a closed densely de ined linea ope a o . Then σp(T)∪
σ (T) = σp(T)∪σp(T0).
P oo . Le z∈σ (T). Then he linea space (T−zI)(X) is no dense in X. By
Hahn–Banach heo em [9] he e exis s y∈X0 {0}such ha ((T−zI)x, y) = 0
and he e o e (Tx, y) = (x, zy) o any x∈X. Hence y∈DT0and T0y=zy.
Thus z∈σp(T0). Le now z∈σp(T0). Then he e exis s y∈DT0 {0}such
ha T0y=zy. Hence (Tx, y)=(x, T0y)=(x, zy) = (zx, y) and he e o e
((T−zI)x, y) = 0 o any x∈DT. So we ha e (T−zI)(DT)⊆ke y. Thus
he se (T−zI)(DT) is no dense in X. I ollows ha z∈σ (T)∪σp(T). ¤
P oposi ion 3. Le Xbe a sepa able e lexi e Banach space and T:DT−→
Xbe a closed densely de ined linea ope a o . Then o any n∈N,σn
p(T)is
an Fσ-se and σc(T)is a Gδ-se .
P oo . Le Xσbe he space Xendowed wi h he weak opology. Since a linea
subspace o a Banach space is closed i and only i i is weakly closed and
is dense i and only i i is weakly dense, we ha e ha Tis a closed densely
BOUNDED LINEAR OPERATORS ON REFLEXIVE BANACH SPACES 137
de ined linea ope a o on he space Xσ. Since closed balls o Xa e me izable
and compac in he weak opology, we ha e ha Xσis a coun able union o
me izable compac se s. Applying Co olla y 1, we ob ain ha σn
p(T) is an
Fσ-se o any n∈N. Co olla y 2 implies ha σp(T0) is an Fσ-se . Acco ding
o P oposi ion 2, σp(T)∪σ (T) = σp(T)∪σp(T0). Hence σp(T)∪σ (T) is an
Fσ-se . Since σ(T) is closed, we ha e ha σc(T) = σ(T) (σp(T)∪σ (T)) is a
Gδ-se . ¤
Rema k 1. Recall ha a Banach space Xis called quasi e lexi e i dim X00/X <
+∞. In pa icula , any e lexi e Banach space is quasi e lexi e. Sligh ly mod-
i ying he i s pa o he p oo , one can see ha P oposi ion 3 emains ue
i e lexi i y is eplaced by quasi e lexi i y.
3. P oo o Theo em 1
We need some addi ional no a ion and auxilia y lemmas.
Le S=S(R2) be he Schwa z space o apidly dec easing in ini ely di e -
en iable unc ions on he plane; le S0be he dual space o he F ´eche space S
(i is usually called he space o Schwa z dis ibu ions [9]) and le Φ : S0−→ S0
be he Fou ie ans o m. Le also α:R2−→ R+be gi en by he o mula
α(x, y) = (1 + x2)−1(1 + y2)−1.
Conside he space E={ ∈S0:α·Φ ∈L2(R2)}endowed wi h he inne
p oduc
( , g)E= (α·Φ , α ·Φg)L2(R2)=ZZ
R2
α2(x, y)Φ (x, y)Φg(x, y)dx dy.
Since he map 7→ α·Φ is a linea homeomo phism o S0on o S0, we ha e
ha Eis a Hilbe space and he opology o Ede ined by he inne p oduc
(·,·)Eis s onge han he opology o S0. No e also ha L2(R2)⊂Eand he
opology o Eis weake han he na u al Hilbe space opology o L2(R2). We
shall also use he ollowing no a ion. Fo wo unc ions Aand Bde ined on he
same se we w i e A¿Bi he e exis s c > 0 such ha |A|⩽c|B|. Fo ϕ∈S
deno e
p(ϕ) = ZZ
R2¯¯¯¯µ1 + ∂4
∂x4¶µ1 + ∂4
∂y4¶ϕ(x, y)¯¯¯¯dx dy.
Clea ly pis a con inuous no m on he locally con ex opological ec o space
S.
Lemma 1. kϕ· kE¿p(ϕ)k kE o ∈Eand ϕ∈S.
P oo . By de ini ion kϕ· kE=kα·Φ(ϕ· )kL2(R2)¿ kα·(Φϕ∗Φ )kL2(R2),
whe e ∗deno es he con olu ion o unc ions. Using he de ini ion o pand
he well-known p ope ies o he Fou ie ans o m, we ob ain Φϕ¿p(ϕ)·β,

138 S. A. SHKARIN and O. G. SMOLYANOV
whe e β(x, y) = (1 + x4)−1(1 + y4)−1. Hence
kϕ· k2
E¿p2(ϕ)kα·(β∗Φ )k2
L2(R2)=p2(ϕ)ZZ
R2
α2(x, y)×
×ZZ
R2
|Φ (u, )|2β(x−u, y − )du d ZZ
R2
|Φ (s, )|2β(x−s, y − )ds d dx dy =
=p2(ϕ)ZZZZ
R4
|Φ (u, )Φ (s, )|ZR
β(x−u, x−s)dx
(1 + x2)2ZR
β(y− , y − )dy
(1 + y2)2dud dsd .
One can easily e i y ha
ZR
dx
(1 + x2)2(1 + (x−u)4)(1 + (x−s)4)¿1
1 + (u−s)4·1
(1 + u2)2+1
(1 + s2)2¸.
Thus
kϕ· k2
E¿p2(ϕ)ZZZZ
R4
|Φ (u, )Φ (s, )|
1+(u−s)4·1
(1 + u2)2+1
(1 + s2)2¸×
×1
1+( − )4·1
(1 + 2)2+1
(1 + 2)2¸du d ds d .
Pe o ming he change o a iables a=u−s,b= − (we pass om a iables
u, , , s o a iables u, , a, b) in he las in eg al and deno ing g=α·Φ , we
ha e
kϕ· k2
E¿p2(ϕ)ZZ
R2

ZZ
R2
|g(u, )g(u−a, −b)|
1 + a4·1+(u−a)2
1 + u2+1 + u2
1+(u−a)2¸×
×1
1 + b4"1+( −b)2
1 + 2+1 + 2
1 + ( −b)2#du d #da db.
Since g∈L2(R2), we ob ain
ZZ
R2
|g(u, )g(u−a, −b)|du d ⩽kgk2
L2(R2)=k k2
E.
Hence,
kϕ· k2
E¿p2(ϕ)k k2
EZZ
R2
1
1 + a4sup
u∈R·1+(u−a)2
1 + u2+1 + u2
1+(u−a)2¸1
1 + b4×
×sup
∈R·1 + ( −b)2
1 + 2+1 + 2
1+( −b)2¸da db ¿p2(ϕ)k k2
EZZ
R2
1
1 + a2
1
1 + b2da db
¿p2(ϕ)k k2
E.
Thus kϕ· kE¿p(ϕ)k kE.¤
BOUNDED LINEAR OPERATORS ON REFLEXIVE BANACH SPACES 139
Le also γ:R2−→ C,γ(x, y) = x+iy and A⊂D={z∈C:|z|⩽1}
be an in ini e locally compac se . Then he e exis s a sequence o compac
se s Kn⊂Csuch ha Knis con ained in he in e io o Kn+1 in A o any
n∈Nand A=S∞
n=1 Kn. Since Ais locally compac , we also ha e ha he se
A Ais compac . Hence o any n∈N he e exis s an in ini ely di e en iable
unc ion ϕn:R2−→ [0,1] wi h bounded suppo such ha ϕn¯¯γ−1(Kn)≡0
and ϕn¯¯γ−1(A A)≡1. Conside he space
EA={ ∈E: supp ⊆γ−1(A) and k kA<+∞},
whe e k k2
A=k k2
E+P∞
n=1 k ·ϕnk2
Eand supp is he suppo o he gene -
alized unc ion . I is s aigh o wa d o e i y ha (EA,k · kA) is a Hilbe
space.
Fo any (x, y)∈R2, he symbol δx,y s ands o he Di ac’s δ- unc ion con-
cen a ed in he poin (x, y). Since he unc ion Φδx,y is bounded, we ha e ha
δx,y ∈E. I (x, y)∈R2is such ha x+iy ∈A, hen he e exis s n∈N o
which x+iy ∈Kn. Hence δx,y ·ϕm= 0 o m⩾n. The e o e
kδx,yk2
A=kδx,yk2
E+Ã1 +
n−1
X
k=1
ϕ2
n(x, y)!<+∞.
I ollows ha δx,y ∈EA o x+iy ∈A. Le us e i y ha he map x+iy 7→ δx,y
is con inuous om A o he Hilbe space EA. Le xn+iyn∈A, (xn, yn∈R),
xn→x,yn→yand x+iy ∈A. Then he e exis s n∈N o which x+iy ∈Kn.
Since Knis con ained in he in e io o Kn+1 in A, we ha e xm+iym∈Kn+1
o su icien ly la ge m. Fo such m,
kδxm,ym−δx,yk2
A=kδxm,ym−δx,yk2
E+
n
X
k=1
kϕk(xm, ym)δxm,ym−ϕk(x, y)δx,yk2
E.
The de ini ion o k·kEand he Lebesgue heo em imply ha kδxm,ym−δx,ykA→
0 as m→+∞. The con inui y o he map x+iy 7→ δx,y is e i ied. Le now
HAbe he closu e in EAo he linea span o he se {δx,y :x+iy ∈A}. Since
he map x+iy 7→ δx,y is con inuous, HAis sepa able as a closed linea span o
a sepa able se . Thus (HA,k · kA) is a sepa able Hilbe space. Conside now
he ope a o T:S0−→ S0de ined by he o mula
T =γ· .
Lemma 2. T(HA)⊆HAand he es ic ion TAo T o HAconside ed as
an ope a o on he Hilbe space HAis bounded. Mo eo e , σ(TA) = A,
σp(TA) = σp,1(TA) = Aand he ope a o TA−zI has dense ange in HA o
any z∈Cwhich is no an isola ed poin o A. Mo eo e , he e exis a cons an
c⩾1and a dec easing con inuous unc ion a: (0,+∞)−→ (0,+∞), which do
no depend on Asuch ha kTAk⩽cand k(TA−zI)−1k⩽a( dis (z, A)) o
z∈C A.
140 S. A. SHKARIN and O. G. SMOLYANOV
P oo . Le η∈S. Acco ding o Lemma 1 o any ∈HA,
kη· k2
A=kη· k2
E+
∞
X
n=1
kη· ·ϕnk2
E
¿p2(η)Ãk k2
E+
∞
X
n=1
k ·ϕnk2
E!
=p2(η)k k2
A.
Choose an in ini ely di e en iable unc ion γ0wi h compac suppo such ha
γ0(x, y) = γ(x, y) o x2+y2⩽1. Since he suppo o any ∈EAis con ained
in he se γ−1(A)⊆ {(x, y)∈R2:x2+y2⩽1}, we ha e T =γ0· and
supp T ⊆supp ⊆γ−1(A) o any ∈EA. F om he ac ha T =γ·
and Lemma 1, i ollows ha kT kA¿p(γ0)k kA o any ∈EA, whe e
pis he no m de ined by a he beginning o Sec ion 3. Hence he e exis s a
cons an c⩾1 such ha kT kA⩽ck kA o any ∈EA. Thus he ope a o
T¯¯EAac s boundedly on he Hilbe space EAand kT¯¯EAk⩽c. Le x, y ∈R
be such ha x+iy ∈A. Then Tδx,y =γ(x, y)δx,y ∈HA. Hence Tmaps he
dense in HAlinea span o he se {δx,y : (x, y)∈γ−1(A)}in o HA. Since he
ope a o T¯¯EAis bounded wi h espec o he no m k · kAand HAis comple e,
we ob ain ha T(HA)⊆HAand kTAk⩽c.
Le ρ∈C∞[0,∞) be such ha ρ¯¯[0,1/2)∪[9c,+∞)≡0, ρ¯¯[1,8c]≡1 and x0, y0∈
Rbe such ha z=x0+iy0∈C Aand |z|⩽2c. Deno e
η(x, y) = 1
x+iy −zρ³|x+iy −z|
dis (z, A)´.
Clea ly ηis an in ini ely di e en iable unc ion wi h bounded suppo and
η¯¯γ−1(A)=1
γ−z¯¯γ−1(A). Since he suppo o any ∈HAis con ained in he
se γ−1(A), we ha e ha η·(TA−zI) = (T−zI)(η· ) = o any ∈HA.
Since T =γ· , he ope a o T−zI is in e ible and k(TA−zI)−1k ¿ p(η).
I is s aigh o wa d o e i y ha p(η)¿( dis (z, A))−6. Hence he e exis s a
cons an c2>0 o which k(TA−zI)−1k⩽c2dis (z, A)−6 o z∈C,|z|⩽2c
and z /∈A. I |z|>2c hen he es ima e kTAk⩽cimplies ha TA−zI is
in e ible and k(TA−zI)−1k⩽2|z|−1⩽4 dis (z, A)−1. Hence σ(TA)⊆A
and k(TA−zI)−1k⩽a( dis (z, A)), whe e a( ) = max{c2 −6,4 −1}. Ob iously,
a: (0,+∞)−→ (0,+∞) is con inuous and dec easing.
No e ha he spec um o he ope a o Tis he en i e complex plane C, is
pu ely poin spec um o mul iplici y 1 and o any z=x+iy ∈C(x, y ∈R)
he one-dimensional space ke (T−zI) is spanned by δx,y. Hence
σp(TA) = σp,1(TA) = {x+iy :x, y ∈Rand δx,y ∈HA}.
Fo x+iy ∈A,δx,y ∈HAby de ini ion o HA. I x+iy /∈A, hen supp δx,y 6⊆ A
and he e o e δx,y /∈EA. Hence δx,y /∈HA. I x+iy ∈A A, hen ϕn(x, y) = 1
o any n∈N. Hence δx,y ·ϕn=δx,y o any n∈Nand he e ms o he
BOUNDED LINEAR OPERATORS ON REFLEXIVE BANACH SPACES 141
se ies om he de ini ion o he no m kδx,ykAa e all equal o he same posi i e
numbe and he e o e he se ies di e ges. Hence δx,y /∈EAand he e o e
δx,y /∈HA. F om he las display we ha e σp(TA) = σp,1(TA) = A. This
equali y oge he wi h he al eady p o en inclusion σ(TA)⊆Aimply ha
σ(TA) = A.
I emains o e i y he densi y in HAo he ange o he ope a o TA−zI
when z=x0+iy0∈Cis no an isola ed poin o A. By de ini ions o TAand HA
we ha e ha δx,y ∈(TA−zI)(HA) o any (x, y)∈γ−1(A) {(x0, y0)}. I z /∈A
we ha e ha {δx,y : (x, y)∈γ−1(A)} ⊂ (TA−zI)(HA). Le z∈A. Since zis
no an isola ed poin o Aand he map x+iy 7→ δx,y om A o HAis con inuous,
we ha e ha δx0,y0is a limi poin o he se ©δx,y : (x, y)∈γ−1(A) {(x0, y0)}ª
in HA. Thus, in any case he se {δx,y : (x, y)∈γ−1(A)}is con ained in he
closu e o (TA−zI)(HA). Hence (TA−zI)(HA) is dense in HAsince he se
{δx,y : (x, y)∈γ−1(A)}has dense linea span HA.¤
Lemma 3. Le K⊂Cbe a nonemp y compac se . Then he e exis s a
bounded linea ope a o CKac ing on a sepa able in ini e dimensional Hilbe
space such ha σ(CK) = σc(CK) = K.
P oo . Le H0=L2[0,1] and T0:H0−→ H0be he classical Vol e a ope a o :
T0 ( ) =
R0
(s)ds. I is well-known ha T0is a bounded linea ope a o and
σ(T0) = σc(T0) = {0}. Le {zn}be a sequence dense in K,Hn=H0 o any
n∈Nand H=
∞
⊕
n=1
Hnbe he Hilbe di ec sum o he Hilbe spaces Hn.
Then His a sepa able Hilbe space. De ine he ope a o CK:H−→ Hby
he o mula (CKx)n= (T0−znI)xn. I is s aigh o wa d o e i y ha his
ope a o sa is ies he desi ed p ope ies. ¤
Lemma 4. The e exis s a bounded linea ope a o T1ac ing on `2such ha
kT1k⩽1,σp(T1) = σ(T1) = σp,1(T1) = {0}and he ange o T1is dense.
P oo . One can easily e i y ha he weigh ed backwa d shi (T1x)n=
xn+1/(n+ 1) sa is ies he desi ed condi ions. ¤
Lemma 5. The e exis a cons an c1>0and a dec easing con inuous unc ion
a1: (0,+∞)−→ (0,+∞)such ha o any non-emp y σ-compac se A⊂
D he e exis s a bounded linea ope a o QAac ing on a sepa able in ini e
dimensional Hilbe space such ha σ(QA) = A,σp(QA) = σp,1(QA) = A,
he ange o he ope a o (QA−zI)is dense o any z∈C,kQAk⩽c1and
k(QA−zI)−1k⩽a1( dis (z, A)) o z∈C A.
P oo . Pick an inc easing sequence Kn(n∈N) o compac se s such ha
A=S∞
n=1 Kn. Le K0=∅and An=Kn Kn−1 o n∈N. Then he se s
Ana e locally compac as open subse s o compac spaces. The se An(as o
any subse o a sepa able me izable se ) can be decomposed as An=Ac
n∪Au
n,
whe e he se Ac
nis ini e o coun able, Au
nis closed in Anand does no ha e