Báo cáo toán học: "On the existence of hyperinvariant subspaces " pot

The main result of this paper is a Theorem on the existence of nontrivial hyperinvariant subspaces for certain operators Theorem 1.1, which extends simultaneously the results of Wermer [

ON THE EXISTENCE OF HYPERINVARIANT SUBSPACES

AHARON ATZMON

0 INTRODUCTION

Throughout this paper, £ will denote an infinite dimensional complex Banach space and £(£) the algebra of all bounded linear operators on £ For an operator Ain Y(£) we shall denote by A* its adjoint acting on the dual space E*, and by (AY its commutant, that is, the set of all operators in Y(E) which commute with A

We recall that a (closed) subspace M c £ is called invariant for an ope- rator A in P(E) if Axe M for every xé M The subspace M is called hyperin- variant for A, if it is invariant for every operator in (4)’ We say that M is not trivial, if M4{0} and M#E

In the sequel, we shall denote by N the set of all positive integers, by Z the set of all integers, by C the set of complex numbers, and by T the unit circle {ze C:\z| = l}

The main result of this paper is a Theorem on the existence of nontrivial hyperinvariant subspaces for certain operators (Theorem 1.1), which extends simultaneously the results of Wermer [23], Sz.-Nagy and Foias [22, p 74], Gellar and Herrero [13], and a recent result of Beauzamy [2]

In general terms, our main result asserts that if A is an operator in #Œ), and there exist sequences (x,),¢z © £ and (Mi)nez c E* with x40 and y,4+0 such that Vane Z

then under some additional conditions, either A is a multiple of the identity ope- rator or A has a non trivial hyperinvariant subspace

An example of such additional conditions (which is a particular case of Theorem J.1(a)) is, that for some integer k > 0

Ixzll + Ilyzil = Odnl“), 2 > zoo

This condition clearly holds, if A is invertible, and there exist non zero vectors

xạ€ £ and ype E* such that

JJ-4'zall + l4*“»;lÌ = O(n), 1 > boo.

In addition of providing a common principle to the results of [23], [22, p 74] and [13], our hypotheses are considerably weaker than theirs, and hold in some cases in which neither of these results is applicable One such example (see Section 6)

is the class of Bishop operators considered by A M Davie in [7]

The contents of this paper are as follows:

In Section 1 we state our main results and some of their consequences

In Section 2 we assemble some preliminary results from harmonic analysis and the theory of analytic vector functions, which are needed in the proof of Theorem 1.1

In Section 3 we present the proofs of our main results stated in Section 1 After proving Theorem 1.1 we deduce from it, by using a theorem of Helson [14, Theorem 3], the result of Wermer [23] Then we prove a general Banach space Lemma which enables us to deduce from Theorem 1.1 the extension of the result

of Sz.-Nagy and Foias [22, p 74] which is given in [6, p 134], and also the following result (which is a particular case of Theorem 1.5):

If E is a Hilbert space and A is an operator in @(E) such that for some vectors

In Section 4 we also extend the results of [l, Theorem 1, and Proposition 6]

In Section 5 we introduce the class of generalized bilateral weighted shifts and extend the results of Gellar and Herrero [13] concerning the existence of non trivial hyperinvariant subspaces for bilateral weighted shifts

In Section 6 we give some examples, and apply the results of Section 5 to certain operators on homogeneous Banach spaces on T We also use these opera- tors to disprove a conjecture of Gellar [12, p 543] We conclude with some com-

ments and problems concerning the existence of invariant subspaces in a certain class of operators which contains the Bishop operators

In considering condition (0.1) we were inspired by the recent paper of Beauzamy [2], although his conditions and methods are different from ours

We wish to express our thanks to Professor Bernard Beauzamy for provid- ing us with preprints of his papers [2] and [3] We also thank Professor Domingo Herrero for several comments concerning Section 6

1 STATEMENT OF MAIN RESULTS

Before stating our main results it will be convenient to introduce the following:

DEFINITION A sequence of real numbers (p,),¢z such that py = 1 and p, 21, Vue Z, will be called a Beurling sequence if the following conditions hold:

We shall also adopt the following convention: We shall say that the sequence

of real numbers (a,),e7, is dominated by the sequence of real numbers (6,),¢7 if there exists a constant c > 0 such that

a, <c-b,, WneZ

An analogous convention will be used for one sided sequences (a,)%.9 and (b,)%.9- THEOREM 1.1 Let A be an operator in LCE) and assume that there exist sequences (X nex cE and (V)nez & &*, with xyA#O and yox¥#0, such that (0.1) holds Vne Z

Then each of the following conditions implies that either A is a multiple of the identity operator or A has a non trivial hyperinvariant subspace:

(a) The sequence (||¥,|))nez i8 dominated by a Beurling sequence and

jor some integer k > 0

(b) The sequence (\\x,|)nez is dominated by a Beurling sequence and

for some integer k > 0

(C) The sequences (|Xall)aex aHđ (||y»Ì„e„ are dominated by Beurling sequences and the union of the singularity sets of the two analytic vector valued functions G, and Gy defined on CNT by:

0

— W y1 >1

:>—co

contains more than one point

(d) x9 is not contained in the closed span in E of the set {x,:neéZ, n#0},

Yo is not contained in the closed span in E* of the set {y,:n€ Z, n#0}, and

(1.7a) X= a -(log*jix,]} -~ logtiy,') < oo

neZ 1 + m

and for some constant b > 0

(17) lXnh Š ĐlXssil and ial S Ð Juái, VneZ

(e) For some integer J

"e2

Condition (c) calls for some explanations As we shall see in Section 2, the assumption that ((}x,/),cz and (lyn!),ez are dominated by Beurling sequences implies that the power series defining G, and G converge absolutely (in the Z norm and E* norm respectively) in their corresponding domains Therefore G, and G, are ana- lytic vector functions in CNT

If G is a vector valued analytic function in C\T, then a singular point of G

is a point 4¢T, which has no neighborhood into which G admits an analytic continuation

Remark It follows from (0.1) that (A z)G,(z) =: X9 for ;Z,l, and there-

fore if A has the single valued extension property (s.v.e.p.), the singularity set

of G, coincides with o,(x9), the local spectrum of xạ with respect to A (For the definition of s.v.e.p and local spectrum see [6, p I].) Similarly, if A* has the s.v.e.p., the singularity set of G, is o4(%) Thus, assuming that the operators A and A*

have the s.v.e.p (which is no loss of generality in considering the existence of hyper- invariant subspaces) we see that part (c) of Theorem 1.1 can be formulated as follows:

(c) The sequenees (|| x„||)„e„, and (IIy„lÌ)„«„ are dominated by Beurling sequences and o,(X%9) Uo4(¥o) contains more than one point

We thank the referee for these observations

An immediate consequence of Theorem 1.1 is:

THEOREM 1.2 Let A be an invertible operator in Y(E) and let xạc E and

yo € E* be non zero vectors If the sequences (A"X»),ez, and (A*"Yo)nez Satisfy one of the hypotheses (a) — (e) of Theorem 1.1, then either A is a multiple of the identity operator or A has a non trivial hyperinvariant subspace

As we shall see in Section 3, Theorem 1.2 implies the following result of

J Wermer:

THEOREM 1.3 (Wermer [23]) Jf A is an invertible operator in LE) then each

of the following two conditions implies that A satisfies the conclusion of Theorem 1.2:

for some integer k 2 0

nez 1 +n

and the spectrum of A contains more than one point

REMARKS 1 An important difference between Theorem 1.2 and Theorem 1.3

is the following: The spectral radius formula implies (see Section 2 or [23]) that the spectrum of an operator which satisfies the hypotheses of Theorem 1.3 is contained

in the unit circle T Similarly all the other known extensions of Wermer’s Theorem (cf [19] or [20, Theorem 6.3]) deal with operators which have a portion of their spectrum (that is, the intersection of the spectrum with some open set in the

plane) contained in a smooth arc On the other hand, no such restrictions on the

spectrum are imposed by the hypotheses of Theorem 1.2 This permits for example

an application of Theorem 1.2 to certain weighted shifts whose spectrum consists

of an anulus (such as the one described in Section 6)

2 The second part of Theorem 1.3 was proved by Wermer in [23] under some- what more restrictive conditions However as shown in the different proofs of Wermer’s Theorem given in [6, p 154] and [1, Section 6], these restrictions are not needed

3 Wermer stated in [23] only the existence of non trivial invariant subspace for A which are also invariant under A~4, but his proof actually produces hyper-

invariant subspaces This fact is explicitely stated and proved in the above men- tioned proofs in [6] and [1]

As we shall show in Section 3, Theorem 1.! also implies the following exten- sion of the result of Sz.-Nagy and Foias [22, p 74] which is given in [6, p 134] THEOREM 1.4 (Colojoara and Foias) Let E be a reflexive Banach space, and (Palnex a increasing sequence of positive numbers such that

mooo Py

for some constant c > 0 and integer k > 0

Let A be an operator in £(E) such that

REMARKS | A simple condition which implies (1.11) with ¢ =: | and k => 01s:

lim sup Paty cy,

If E is a Hilbert space, one can replace condition (1.12) by a weaker condition which does not impose restrictions on the norms of the operators A”, ne N More precisely we have the following result:

THEOREM 1.5 Let £ be a complex Hilbert space and (Pj)nen @ Sequence which satisfies the hypotheses of Theorem 1.4 Let A be an operator in Y(E) and assume that there exist vectors xe E and ye E* such that (1.13) and (1.14) are satisfied

and that

(1.15) supf{llpz 1x L4?” A"x||:m, n e N} < 00

and

(1.16) sup{||pa'p, ‘A A*"yl|: m, ne N} < co

Then the conclusion of Theorem 1.4 holds for A

Evidently, (1.12) implies (1.15) and (1.16) but not conversely

Another consequence of Theorem 1.1 is an extension (Theorem 3.6) of the

In what follows we shall denote by C(T) the set of all complex continuous functions on T For fe C(T) and »éZ we denote by Ẩm) the #-th Fourier coefficient of f that is

2z

Âm == \ fede im 2n

9

+ We shall require the following:

LEMMA 2.1, Let (6,)ne7 b¢ a sequence of real numbers such that o,21, Wne Z and assume that

néZ 1 +

and that for some constant e > 0

Then for every 0 < a <b < 2k, there exists a function f#0 in C(T) which is sup- ported by the arc

2 Lemma 2.1, even without the assumption (2.2), appears (in equivalent form) in [13, Lemma 3] However the proof given there is not correct, since in the estimates of the Fourier coefficients in [13, p 180], the third inequality holds only

if the sequence (7) (the sequence ø„ in the notation there) is eventually

neN decreasing This assumption in conjunction with (2.1) is stronger than (2.2)

We do not know whether or not the conclusion of Lemma 2.1 is true without assumption (2.2)

Proof of Lemma 2.1 Let B, = logo,, a ¢ Z, and consider the piecewise linear function @ on ( -00, oo) which satisfies p(n) == 8, for ne Z

It is easy to verify that (2.1) implies that

Using the fact that

\ lÊ(n — ĐIexp(e(n — Ð)) di — IÊ(x)Iexp(ø(3))dx

Let now f be the function in C(T) defined by

fe") =eg(t), O<t < 2n

We claim that f has the required properties Indeed

fn) = -=—= —#u —a), VneZ

and therefore (2.3) follows from (2.8) The assumptions on g imply that {#0 and that f is supported by I’ This completes the proof of the lemma

Throughout the rest of this section p = (p,)nez Will be a Beurling sequence and A, will denote the set of all functions f in C(T) such that & | flo, < © Since p, > 1, Vn Z, it follows that È) LẦU) < co for fe 2o, and therefore

neZ

the Fourier series of f converges uniformly on T to f:

Jt is well known, and easy to verify, that (1.1) implies that, with norm

l= Lt fien, fe Ap,

A, is a Banach algebra with respect to pointwise addition and multiplication of functions on T It is also clear that for every f in A, the sequence of trigonometric polynomials

s(e") = ¥ flue’, neN fink

converges to fin the norm of A,.

It is known (cf [11], p 128) that (1.1) implies that the limits

R, = lim pl” and R,:= limp} n n

>> — CO n~oo

exist This fact in conjunction with (1.2) implies that R, =: R == 1 Therefore by (11, p 130], the maximal ideal space of A, can be identified in the natural way with T

REMARKS | The fact that Ñ; 8; = 1 implies that, if (x,),¢7 iS a sequence

of vectors in a Banach space E and (;x„j)„ez is dominated by a Beurling sequence then

limsup]x„:"* < l and limsupix,!"" < 1

Consequently the functions G, and G, in (1.5) and (1.6) are analytic in C>.T

2 It also follows from (1.5) and (1.6) and the previous remark that

for every fin A, and S in A®

Let 7 denote the Banach space of all complex sequences (¢,),ez for which the norm

iat) ear em Hel] == sup -""' nEZ ĐT,

is finite

It is easily verified that the mapping

S+(Srez, SeAs

establishes an isometric isomorphism between A¥ and ¢?

For every fe A, and S¢ A* we shall denote by f-S the element of A> which

A(T) is called the hull (or co-spectrum) of J

Since the Banach algebra A, is regular, every element S in A} has a well defined support (see [17], p 230), which is the complement (with respect to T) of the largest open (in the topology of T) subset U < T, such that <f, S> = 0 for every function f in A, whose support is contained in U

We shall denote the support of Se A* by Z(S) It is clear that 2(S) is empty if and only if S = 0, and that V fe A, and V Se AF

Z(-S) c š(S) n support()

Ín the sequel we shall require the following:

Lemma 2.2 Let Se A*® and let J be the ideal in A, which consists of all functions f¢ A, such that f-S = 0 Then h(J) = Z(S)

Proof We show first that h(J) ¢ X(S) Suppose that Ae T\23(S), and let L

be an open arc on T which contains A and is disjoint from 2(S) Let f be a function

in A, which is supported by ZL, and f(A) = 1 It follows from the definition of 2(S) that f-S = 0, and therefore fe J Since f(4) = 1 we deduce that A¢A(/) This shows that A(J) < 2(S)

To prove that Z(S) < A(J), consider Ae T\A(J) There exists a function f

in A, such that f-S = 0 and f(A) = 1 Let I be an open arc on T, which contains

A, such that |f(z)| = 1/2, Vzel Let J be the principal ideal generated (algebra- ically) in A, by f It follows from [17, Corollary 5.7, p 224] that J contains every function g in A, which is supported by I That is for every such function g, there exists a function @ ¢ A, such that g = @-f, and therefore, since f-S = 0, we have

We shall denote the set of singular points of 5 by sing(S)

In the proofs of Theorem 1.1 (c) and Theorem 4.1, the following result will

be of fundamental importance:

LEMMA 2.3 For every Sin A*, 3(S) = sing(S)

The idea of this result, in the setting of Fourier transforms (at least for Beurling sequences of polynomial growth) goes back to the work of T Carleman [5, Ch 1], (see also [17], p 179)

For general Beurling sequences this result is essentially contained in [9] Since it is not explicitely stated there, we include a proof

Proof of Lemma 2.3 Let S¢ A}, and consider the ideal J associated with

S as in Lemma 2.2 It follows from Lemma 2.2 and [9, Theorem 2.4 and Example 3.1] that sing(S) < 2(S), and from Lemma 2.2 and [9, Theorem 8.1 and the example which follows] that 2(S) < sing(S)

in the proof of Theorem 1.1 we shall also require the following result:

LEMMA 2.4 Let E be a complex Banach space, and let F and G be functions with values in E and E* respectively, defined and analytic in CX Assume that there exists an open disc D, with center on T, such that (xe E and Wye E* the complex functions

z— (F(z),y> and z—<x,G(z)), zeCX\T can be continued analytically into D Then F and G admit analytic continuations into D

Proof Let D, be an open disc whose closure is contained in D Remembering that a (complex) analytic function which is analytic in a neighborhood of a closed disc satisfies Lipschitz condition (of order 1) on that disc, we obtain from the hypotheses that Vy e E*

sup{|(F(z2), ¥> — <F(&), ¥>\ + \Z2 — Za]74: %, Z22€ DINT, 2#z¿} < 00 Therefore by the uniform boundedness principle

sup{|| F(z) — F(z)||-1z2 — 2,)72! 2., 22€ DiNT, Zz¡#z¿} < co

Consequently, F is uniformly continuous on D,\T and therefore admits a conti- nuous extension to D, Since D, is an arbitrary open disc whose closure is contained

in D, we conclude that F admits a continuous extension to D, which we denote

by F, From the hypotheses of the lemma it follows that Vye E*, the complex function

3 PROOFS OF MAIN RESULTS

We begin by introducing a notation which will be used throughout this section Let A be an operator in ¥(E) and assume that (x,),¢7 CE and (),),ez CE* are sequences such that (0.1) holds

For functions f and g in C(T) such that

(3.1) néZ ¥ AMI xl] < co and ¥ [Bal Ily,[| < 00 néEZ

we shall denote by u(f) and v(g) the vectors in FE and E*, respectively, defined by

uf) = 3 f(n)x„, and o(g)= ¥ a(n)yy nez "nez

Since E and E* are Banach spaces it follows from (3.1) that the series defining u(f} and v(g) converge in the respective norms

In the proof of Theorem 1.1 we shall require the following:

LEMMA 3.1 Let A be an operator in P(E), and let (X,)nez CE and (Yy)ncz = E*

be sequences such that (0.1) holds Assume that f and g are functions in C(T) which

satisfy (3.1) Then VW Be(A)’

Proof We show first that

(3.5) (Bx, Vo = (Bxjin, Yo, WBE(A), VikeEZ

Assume first that & is a nonnegative integer Then by (0.1)

A*x;,=x;x.,, WieZ and A**‘yy = ÿy

Therefore V Be (A)’ and Vje Z

CBx;, yụ) = (Bx;, A*kyg) ==: &A*Bx,, Yoo =

= (BA*x;, Yo) = CBX¡+¡, ViỀ

Suppose next that k is a negative integer Then by (0.1),

A-*xj4,=%;, Wie Z and A*-*y, = yy

Therefore V Be(A)’ and VjeZ

(BX; 44> Yoo = (Bxj44, A* "yy =

~= ‹A-*BX/j.e, Vuồ = <BA-*xj., 12) = CBX/, My}

Thus (3.5) is proved

From (3.1) and (3.5) we deduce that VW Be (A)’,

YA DEAK BX 4250! = DL AVE Bx), HDI <

< HBICE AD!) -CE BO! nD < 00

Thus noticing that

ƒ-gœ = % few, YneZ

we obtain in particular that (3.2) holds By changing the order of summation (which

is permitted by virtue of the absolute convergence of the series above) and using (3.5) once again, we obtain that V Be (A)’

<Bu(f), v(g)> = ,à,ÂUÊ0)CB, M=

= x FU) "¡„GZ2.Jtk:sn BX,, Yo) = yy Zˆ20)(Bx„, xù

This completes the proof of the lemma

In the sequel we shall also need the following:

LEMMA 3.2 Let A be an operator in Y(E) and assume that there exist non zero vectors ué E and vé E* such that

Then A has a non trivial hyperinvariant subspace

Proof Let M be the closure in E of the linear manifold {Bu: Be (A)’} It is clear that M is a hyperinvariant subspace for A M#{0} since ue M,-and since v0, it follows from (3 9c that M4E Thus M is not trivial, and the lemma is proved

REMARK Using the Hahn-Banach Theorem it is easy to show that the hypotheses of Lemma 3.2 are also necessary for the existence of a non trivial hyperinvariant subspace for A

Proof of Theorem 1.1 First we notice that if AO and A is not injective, then ker(A) is a non trivial hyperinvariant subspace for A, and if A* is not injective then the closure of the range of A is a non trivial hyperinvariant subspace for A Thus in what follows we shall assume that A and A* are injective This assumption,

in conjunction with (0.1) and the hypothesis that x40 and yạ#0, Implies that

Proof of (e) From (3.5) we deduce that

<Bx,, yo> = CBx,+,, y_„», WBE(A)', VneZ and therefore V Be (A)’,

|< Bx;, Yor| < ||B Iinf Iixus,| l7_„Í- Consequently, (1.8) implies that

2.- 1511

Since })#0 and by (3.7) also x;#0, we obtain from (3.8) and Lemma 3.2 that

A has a non trivial hyperinvariant subspace

Proof of (d) Consider the sequence

6, = max{'x, |, 1}-max{,y,", I}, ae Z

It follows from (1.7a) that the sequence (¢,),<7 satisfies (2.1), and from (0.1) and (1.7b) we obtain that it also satisfies (2.2) with ¢ = i|A|[? + 6? + 1

Let F, and ©, be disjoint open arcs on T By Lemma 2.1 there exist functions f#0 and g#0 in C(T), supported by I, and I, respectively, such that

> LÑn)iơ, <cœ and DE lg(n)'o, < ©

Noticing that max{x„il, (|pzl} < ơ,, Vớe Z2, we obtain that (3.1) holds for f and g Since I’, and I, are disjoint, f-g == 0, and therefore by Lemma 3.1 also (3.4) holds Thus by virtue of Lemma 3.2 the assertion of the theorem will follow, if we show that u(f)#0 and v(g)<0

Since xy is not in the closed span of the set {x,:neZ, nAO}, we deduce from (0.1) that for every negative integer p, x, is not in the closed span of the set {x,:né Z, n#p} Since f= 0 on I,, and f#0, there exists a negative integer q such that f(g) #0 (see [17], p 90, Corollary 3.14) Combining these facts we obtain that

Ix, # — LIM, necz

on T Therefore the hypotheses of (c) imply that there exist 2,, 4.€T, A, #A such that 4, is a singular point of G, and 2, is a singular point of G, By Lemma 2.4 there exist vectors x € E and y ¢ E* such that A, is a singular point of the function

z Gz), y>, zc CNT and A, is a singular point of the function

z— (x, Gz), ze CNYT.

Therefore, if S, and S, are the elements in A¥ defined by

Š@)=(x.„,y> and $,(n) = <x, y_,), VneZ,

we deduce from Lemma 2.3 that 4, € 2(S,) and A, € 2(S))

Let [, and I, be two disjoint open arcs on T such that 4, el, and A, € Fg Since 1, € Z(S,) and A, € Z(S,), there exist functions f and g in A,, supported by I, and Ƒ; respectively, such that ¢f, S,>#0 and <g, S.540 But a simple computa- tion shows that

h Sy> = &u() 1 and &, S;> = «Xx, 0(8)>,

and therefore u(f)40 and v(g)40 This completes the proof of (c)

Proofs of (a) and (6) If the hypotheses of (a) or (b) are satisfied and one of the functions G, or G,, defined by (1.5) and (1.6), has more than one singularity, the conclusion of the theorem follows from part (c) (since ((1 + |#!)*),¢z is a Beurling sequence) Remembering that each of these functions has at least one singu- larity on T, we see that it suffices to consider the case in which each of them has exactly one singularity on T

We shall show first that if G, has a single singularity at 4g « T and (1.3) holds, then

This will prove the assertion, since (3.9) implies that either 4 = 2¿ƒ or ker(A — Aol)

is a non trivial hyperinvariant subspace for A

It suffices to prove (3.9) in the case that A) = 1, since the general case can be deduced from this one, by replacing the operator A by A714 and the sequences (xXz)„e„ and (¥,),ez by the sequences (A3x,),c7 and (Azy,),cz- (It is easily verified that these replacements preserve all the hypotheses.)

Thus we assume that z = | is the only singularity of G, and that (1.3) holds, and we shail show that

For this we introduce the difference operator A defined on sequences (a,)°°.y ¢ E by

Aa, = @,—4,-1, meEN, and Ady = a

Tf (a,)%°.9 ¢ Eand F is the (formal) power series F(z) = Š q„z”, it is easy to

Now if G, has a single singularity at z =- 1 and (1.3) holds, then according

to [15, Theorem 3.12.7, p 60], G, is a polynomial in (1 — z)7! of degree not exceeding & (with coefficients in £) Therefore the same is true for the function

A(z) =: 271G (274), ze C\{1}

and consequently the function p(z) = (1 — 2)*+1A(z) is a polynomial in z (with coefficients in E) of degree at most k From (0.1) and the definition of G, we obtain that

It is easily verified that

AH A'Xy) =2 AP-HA — Dixy, Wa >j and therefore (3.10) follows by setting » =A ~-1 in (3.12) Thus (a) is proved

A similar argument shows that if G, has a single singularity at 4,¢T and (1.4) holds, then (A* —- 4,)**19 + 0 Therefore either A =: AU, or the closure of the range of A ~- 4,7 is a non trivial hyperinvariant subspace for A This proves (b), and completes the proof of Theorem 1.1

Proof of Theorem 1.3 Assume first that A satisfies (1.9) Then for every two non zero vectors x) € E and yy € E*, the sequences (A"Xg)nez and (A*")o)nez, satisfy the hypotheses of part (a) of Theorem 1.2 and the assertion follows

To prove the second part of the theorem, assume that (1.10) is satisfied and consider the sequence p, =: '|A"", ae Z It follows from (1.10) and the fact that Atta <A A", Vin,ne Z, that p = (p,),¢z 18 a Beurling sequence

Since Vx ¢ £ and Wy ¢ E* we have that

ÏA'xl< pjxl, and [Ay < pilivi, WaeZ the assertion will follow from part (c) of Theorem 1.2, once we show that there exist non zero vectors x)¢ E and y)¢ £* such that the union of the singularity sets of the functions G, and G, associated with the sequences (A’xo),¢z and L4” wa)sez by (1.5) and (1.6) contains more than one point.

To show this, let (A) denote the spectrum of A and consider the resolvent

Assume now that 4, € a(A) By a Theorem of Helson [14, Theorem 3], there exists

a vector Xx) € & such that the vector function

z> R(A, z)x, |z| < 1 has no analytic continuation to any neighborhood of 4, Clearly x,#0 Therefore (3.13) implies that A, is a singular point of the function G, associated with the sequence (A’Xy)nez by (1.5)

By the hypotheses, o(A) contains more than one point, and therefore there exists A, € o(A) such that 1,42, Remembering that o(A) = ø(4*), we obtain by replacing in the above argument 4, and A by 2; and 4*, that there exists a non zero vector Yy € E*, such that A, is a singular point of the function G, associated with the sequence (A*")9),¢z by (1.6) This completes the proof of the theorem The link between Theorem 1.1 and Theorems 1.4, 1.5 and 1.6 is established

by means of the following results:

Lemma 3.3 Let A be an injective operator in &Y(E) and assume that there exists @ sequence (W,),nen C E* and a vector xe such that