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
(0 1) AX, = Xnt2 and A*y, ;= Vat
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
Trang 2In 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 x and y in E,
limsup!4"x;¿ >0, limsup.A4”*y; > 0,
R200 RCO
sup JJ4*"A"xi< co and sup J4"A**yp| < co,
m,nGN n,nGïN
then either A is a multiple of the identity operator or A has a non trivial hyperin- variant subspace
This result clearly extends the result of (22, p 74] and does not impose any conditions on the norms of the operators A”, ne N
We conclude Section 3 by proving Theorem 3.6 which extends a result of Beauzamy [2]
In Section 4 we apply the methods of Section 3 to prove some additional results One such result (which is a particular case of Theorem 4.1) is the following: ff A is a contraction in LCE) such that the intersection of its spectrum with the unit circle T is countable, and the sequence (A"),¢x~ does not converge strongly to the zero operator, then A* has an eigenvalue Consequently, if A is not a mul- tiple of the identity operator it posseses a non trivial hyperinvariant subspace 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
Trang 3com-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:
(1.1) Pinta < Đmn › Vn, ne +,
|
(1.2) 108 Pn so
nez I + nầ
Similarly we define one sided Beurling sequences (p,)§So by replacing in the above definition, Z by N
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
(1.3) llx;ll = On|), 2 :keo
jor some integer k > 0
(b) The sequence (\\x,|)nez is dominated by a Beurling sequence and
(1.4) llz„ll = Oa), n => +00
Trang 4(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:
oo Y xen, zi <i (1.5) Gz) = | — x Xo" 4, lZ¡ >] and oO ¥ vu z'< { asl (1.6) G.(z) = | 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
(1.8) inf! 'Xnajl Wve aii ‘ = 0
"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-
Trang 5have 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:
(1.9) ll4”ll= O(lzi), 12 + 4:00
for some integer k 2 0
(1.10) -eslL 1 < 00
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
Trang 6hyper-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
(1.1) limsup ““**“ < cnt, VneN
mooo Py
for some constant c > 0 and integer k > 0 Let A be an operator in £(E) such that
(1.12) NA" = O(p,), 1 - 00
and assume that there exist vectors x ¢ E and ye E* such that (1.13) lim sup'lp>14"x|j > 0
RAC
and
(1.14) lim sup|'p71A*"yil > 0
HCO
Then either A is a multiple of the identity operator or A has a non trivial hyper- invariant subspace
REMARKS | A simple condition which implies (1.11) with ¢ =: | and k => 01s: lim sup Paty cy,
Tì— CO Pn
This holds in the examples: p, == n/ for some j 2 0; p, =: exp(n’), for some 0<2<15 p, = exp _——— } for some 0 < < co
2 In Section 4 (Theorem 4.5) we extend Theorem 1.4 by showing that the right hand side of (1.11) can be replaced by Kexp(cn™*), for some constants K > 0 ande > 0
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:
Trang 7and 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
following result: ,
THEOREM 1.6 (Beauzamy [2]) Let A be an operator in LCE) such that || Al| = 1, and assume that for some vector xe E
(.17) lim supi) A"x|| > 0
n> oO
Suppose that there exists a sequence of vectors (u,)o.9 C E with tg#0 such that (lu, |[)2.9 is dominated by a (one-sided) Beurling sequence and
(1.18) Au, = m3, VneN
Then either A is a multiple of the identity operator or A has a non trivial hyper- invariant subspace
2 PRELIMINARIES
In this section we assemble some background material from harmonic ana- lysis and the theory of analytic vector functions which will be needed in the sequel Although all of these results are known, some of them do not seem to be readily available in the literature
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
(2.1) logơ,
Trang 8and that for some constant e > 0
(2.2) C710, S Ona, S CO,, VneZ
Then for every 0 < a <b < 2k, there exists a function f#0 in C(T) which is sup- ported by the arc
[= {zéT:a < argz < bY
and which satisfies
(2.3) néZ@ ¥ IẪn)lø, < co
Remarks 1 Under the additional assumption that (6,),¢7 is a Beurling sequence the above result is well known and follows from the Paley-Wiener Theorem (cf [6, p 149], or [8])
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
1 :
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
(2.4) | PO) gy <p
and (2.2) implies that for every —-co < t < co
(2.5) sup ip(x -+ t) — @(X), < 00,
—¬CO<x<œ
therefore, it follows from [4, Theorem i] that there exists a continuous noniden- tically zero function g on (—oo, oo) which is supported by (a, 6), whose Fourier transform g satisfies
(2.6) g(x); exp(9(x)) dx < co
Trang 9Using the fact that
\ lÊ(n — ĐIexp(e(n — Ð)) di — IÊ(x)Iexp(ø(3))dx
0 ~œ
we obtain from Fubini’s Theorem and (2.6) that for some 0 < « < J,
(2.7) 3, lÊ(n — 2) | exp(o(n — a) < 00
It follows from (2.5) and the definition of @ that there exists a constant d > 0 such that
B, =o) < ep(a—a+d, VneZ and therefore by (2.7)
(2.8) „eZ 3 lÊ(n — #)|ø„ < œ 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
Trang 10It 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
n> Qœ 1-900
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
Jim {G(2j-=0, j= 1,2 2-0
and therefore by Liouville’s Theorem (cf [[5], p 100) and the fact that x)#0 and }'9#0, we see that each of the functions G, and G, has at least one singularity on T
By virtue of Lemma 2.1, condition (1.2) implies that the algebra A, is regular, that is for every closed set K c= T and 2e TK there exists a function f in A, such that f(z) =.0 for ze K and ƒ{2) = Ì (see also [8] where a more general result is proved)
For every S in A} (the dual of A,) we set
Sn) ase, S) ned
A simple computation shows that
(7, S> = Y Đu)j§(G—-n) nEZ
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,
Trang 11It is easily verified that the mapping
S+(Srez, SeAs
establishes an isometric isomorphism between AƠ and Â?
For every fe A, and S¢ A* we shall denote by f-S the element of A> which is defined by
{af S> = (fg, S, ge A,
It is easy to verify that V fe A, and VW Se AF
o~ a as 4
ƒ- Sự)= À fín —j)§U), VaeZ
J
For an ideal J c A, we shall use the notation h() ={z<T:ƒ() =0, Vƒe]) 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)
Trang 12that
<8, S> = (9, f-S> =: 0
This shows that I is disjoint from Z(S), hence 7¢2(S) Thus 2(S) ¢ A(J), and the lemma is proved
DEFINITION The Carleman transform of S in 47 1s the function S defined
on C\T by > Sw27, \zi<1 ~ nel S(z) == 0 A — % S(z"”1, lzi>] J„j-›—oo
Since the sequence (SQ) rez is clearly dominated by the Beurling sequence (P-wWnez> it follows (from Remark 1, following the definition of A,) that S is well defined and analytic in CXT
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
Trang 13Proof 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
z —> <h©), 1; z€ (CNT) U D
is analytic in D, and therefore (see also [15], p 53) F, is an analytic continuation of F into D
A similar argument (see also the proof of Theorem 3.9.1 in [15]) shows that G also admits an analytic continuation into D
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:
Trang 14satisfy (3.1) Then VW Be(A)’
(3.2) x | fe(n)< BX, Yoo’ < 00
and
(3.3) (Buf), v(g)> = ZS aces, Đề
Thus in particular, if ƒ-g —= 0, then
(3.4) €Bu(7) t(g)> =0, WBe(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 <
IKE JREZ
< HBICE AD!) -CE BO! nD < 00
Thus noticing that
a
Trang 15we 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
(3.6) (Bu, v) =0, WBe(Ay
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
(3.7) x,#0 and y,40, WredZ
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
(3.8) <Bx;,yạỳ =0, VBe (4)
Trang 16Since })#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, < ©
nez "neZ
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
nạ
and consequently u(f)#0 A similar argument shows that v(g)40, and the proof of (d) is complete
Proof of (c) Since the product of two Beurling sequences is again a Beurling sequence which dominates both, we may assume that ({lx,[)nez and ('¥a lez are dominated by the same Beurling sequence p = (/,),e¢z - Clearly (3.1) holds V fig € A, Therefore by virtue of Lemma 3.1 and Lemma 3.2, it suffices to show that there exist functions fand gin A,, with disjoint supports, such that u(f)#0 and v(g)#0 For this consider the two vector functions G, and G, defined by (1.5) and (1.6) As noted in Section 2, each of these functions has at least one singularity 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
Trang 17Therefore, 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
(3.9) (A4 — Asl)#+1xạ = 0
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
(3.10) (A — Dk +2x, = 0
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
„x0
prove by induction that
G.11) (1 — Đ/F@) = Ÿ (Ala,)2", WjeN
ne=0
Trang 18Now 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 A(z) = YA %)2", jz <1 nrs0 and therefore by (3.11) p(Œ) = W A*+MNA )Z", 2z: <1, „z0
Since p is a polynomial of degree not exceeding & we conclude that (3.12) AR* A’ xg) = 0, Wa > k
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
Trang 19To show this, let (A) denote the spectrum of A and consider the resolvent R(A, z) = (A — 21)", ze C\o(A)
Remembering that ø(4) c T by (1.10) (see the remarks in Section 1) we obtain that
R(A,z) = Y Art" zh < n=l and 0 R(4,2)=— W A12“ zh > 1 Nes QÔ
Therefore If xe E, and Œ; is the function associated with the sequence (4” XÌheZ
by (1.5), we see that
(3.13) G2) = R(4,z)x, Vze€CST,
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
(3.14) sup{||A*"w, |]: m, ae N, m <n} < co and
(3.15) lim sup|4”x, w„à| > 0
Then there exists a norm bounded sequence (v,)°.9 < E* with vyg40 such that
Trang 20Lf in addition there exists a sequence of positive numbers (G,)nen Such that (3.17) limsupjjA*"*"w, 11 <Â,, WmeN
nơo
then
(3.18) jA*" on Sdn, WmeN
Proof tt follows from (3.15) that there exists a number 6 > 0 and an increas- ing sequence of integers (;);ew Such that
(3.19) Kx, Aw, > >d, VkeN,
Since a bounded set in £* is precompact in the w*-topology we obtain from (3.14) that there exists a subnet (A*’w,), <,, of (Aen ken (where A is a directed subset of N) which converges in the w*-topology to some vector vy ¢ E* It follows from (3.19) that léx, rọ)' > ô, and therefore ty #0
For every n¢N consider the subset of E*
Vy = {A??T"w,:ỳ 8 Á, y > ah
By (3.14) V, is bounded and therefore has a w*-limit point v,¢ £* Using (3.14) once again, we see that the sequence (v,)” 9 is norm bounded
From the definition of V,, and the fact the adjoint of an operator in #(£) is continuous with respect to the w*-topology of E*, we deduce that A*”v, == vo, WaneN and therefore
A*"™- Av, — tai) =0, WaeN
Since A* is injective this implies (3.16)
If (3.7) holds, then (3.18) follows directly from the definition of vg This completes the proof of the Jemma
COROLLARY 3.4 Let A be an injective operator in LCE) and assume that there exists an increasing sequence of positive numbers (D,)yen and a vector x © E such that
3.20) ({A"|| = O(p,), 27 00
and
(3.21) limsup¡j|p; 14”xi| > 0
H¬oœ
Then there exists a norm boundedl sequence (bạ)pọ C E° wÍth tạ#O such that (3.16) hoids
If in addition there exists a sequence of positive numbers (q,)nex such that
(3.22) lim sup “f5 < Sđ„, VmeN,
ROO n
Trang 21Proof By the Hahn-Banach Theorem there exists a sequence of unit vectors @Jnen & £* such that
(Pn tA"x, 2) = ||pz!A"x||, Wane N
Thus (3.20) and (3.21) imply (3.14) ana (3.15) with w, = p74z,, ne N
If (3.22) is satisfied, then it is easily seen that (3.17) also holds for the sequence (¥,)nen- Hence the Corollary follows from Lemma 3.3
Coro.iary 3.5 Let E be a Hilbert space, A an injective operator in £(E), and CPz)aen tt increasing sequence of positive numbers which satisfies (3.22) for some sequence of positive numbers (4,),en Assume that there exists a vector xe E such that (3.21) holds and that
(3.23) sup{||pz1 pz, 1A*"A"x||: m, ne N} < oo
Then there exists a norm bounded sequence (v,).) < E with vy#O0 such that (3.16) and (3.18) hold
Proof The Corollary follows from Lemma 3.3, by observing that the assump- tions imply that the sequence w, = p,2A"x, néN satisfies all the hypotheses of the lemma
Proof of Theorem 1.4 By the remarks in the beginning of the proof of Theorem 1.1, we may assume that A and A* are injective Thus applying Corollary 3.4, we deduce from (1.11), (1.12), (1.13), that there exist a norm bounded sequence (v,)72.9 < £* with vy#0 such that (3.16) holds and
(3.24) ||A*"vg|| = O(n), n> oc Consider the sequence (y,),cez < £* defined by
}ạ==Ð_„, n<0; yy¿= A tu, n> 0
It follows from the properties of (ø;)z.; and (3.24 that (0.1) and (1.4) hold for the sequence ;);ez.-
Since £ is reflexive we obtain by a similar argument, using (1.14), that there exists a sequence (x,),ez © E with x)%0 such that (0.1) and (1.3) hold
Thus A satisfies the hypotheses of part (a) 4and part (b)) of Theorem 1.1 and the desired conclusion follows
Proof of Theorem 1.5 By an argument similar to that in the proof of Theorem 1.4 we obtain from the hypotheses and Corollary 3.5 that A satisfies the hypotheses of part (a) of Theorem 1.1, and the assertion follows
Trang 22THEOREM 3.6 Let A be an operator in ¥(E) and let (p,)\°.» be an increasing Beurling sequence which satisfies (1.11) and (1.12) Assume that there exists a vector x€ E such that (1.13) holds, and a sequence (u,)° ¢ E which satisfies the hypotheses of Theorem 1.6 Then the conclusion of Theoremt 1.6 holds for A
Proof Using Corollary 3.4 we obtain from (1.11), (1.12) and (1.13) that there exists a norm bounded sequence (v,)% 5 < E* which satisfies (3.16) and (3.18) with qq va nh, neN
Consider the sequences (x,),¢z ¢ £ and (y,),ez © E* defined by:
Nyt Uln, A< 0; An = Aua, n>Ũ
and
Vn = Vln, US 0; dn == AP" by, neo
It follows from the properties of the sequences (w,)%°.9 and (v,)%.9, and the fact that (p,)% » is a Beurling sequence, that the hypotheses of part (b) of Theorem 1.1 are satisfied for the operator A and the sequences (x,),ez and (),),¢z- This completes the proof
4 ADDITIONAL RESULTS
This section contains some additional results which can be proved by the methods of the previous section
We begin with an extension of the result mentioned in the end of the intro- duction
THEOREM 4.1 Let E be a complex Banach space and let A be an operator in YE) with spectrum o(A) Assume that there exists an increasing sequence of positive numbers (D,nen and a vector x ¢ E, such that conditions (1.11), (1.12) and (1.13) hold If o(A) NT is countable, then A* has an eigenvalue, and consequently, either A is a multiple of the identity operator, or A has a non trivial hyperinvariant subspace Proof First we recall that if A* has an eigenvalue 4 and A#/J then the closure of the range of A — J/ is a non trivial hyperinvariant subspace for A Thus the second assertion of the theorem follows from the first
If A* is not injective then 4 :-: 0 is an eigenvalue of A* Thus in what follows we shall assume that A* is injective We shall show that in this case A* has an eigen- value in T
Trang 23We claim that the singularity set of G, is included in o(4)N T Using (0.1), a simple computation with power series shows that
(A® — zI)G,(z)= yp Wze C\T and therefore
(4.1) G(z) = R(A*,z)y, Wze C\(Tuo(A))
where R(A*, z) denotes the resolvent of A* Since z— R(A*,z)yạ, ze C\a(A*)
is an analytic (E* valued) function, we deduce from (4.1) that T\o(A*) is disjoint from the singularity set of G,, and since o(A) = o(A*), the claim is proved
Consequently, since øơ(⁄) ñ T is countable, also the singularity set of G, is countable, and since it 1s clearly closed and not empty (see the remarks in Section 2) it has an isolated point 4 We shall show that J is an eigenvalue of A*
By Lemma 2.4 there exists a vector x’ ¢ FE such that / is a singular point of the function
z—> (x', G,(z)>, ze CNT
Consider the Beurling sequence p = ((1 + |n|)*), cz (where k is the integer in (1.11)) and let S be the element of A¥ defined by
S(n) =O ,yvp, HeZ Noticing that by (1.6)
S(z) => Xx’, G.(z)>, Ze C\T
we deduce that 1 € sing(Š) and therefore by Lemma 2.3, 2 e š(S)
Let F be an open arc on T which contains 2 but no other -singular point of G, Since A € Z(S) there exists a function f in A, which is supported by I’, such that Cf, S>#0
Consider the sequence (y)),~z © E* defined by y= ve mf), neZ; (we use here the notation introduced in Section 3) that is,
(4.2) y= Yfutdy, neZ JEEZ
Trang 24second part of (0.1), we obtain from (4.2) that
(4.3) A2y,= vì, VneZ
Since (1 + ‘n')*), <7 is a Beurling sequence we obtain that (y;)„e„ satisfes (1.4) Let G be the function associated with the sequence ();,),¢z by (1.6) We claim that A is the single singular point of G For this, consider for every te E the elements S, and L, of Af defined by
(4.4) Sn) = (ty, neZ
and
(4.5) DẤU) =: Cf,ÿ „3, ned
{t follows from (4.2), (4.4) and (4.5) that
A “~
tín) =ƒ/- SÁ(n), Vne2, VieE and therefore
{4.6) L,=f-S,, Wtek
Observing that Vre E
S,(z) = (t, Gz), ze CX\T
we deduce that sing(5,) is included in the singularity set of G., Vt¢ E, and therefore by Lemma 2.3, 2(.S,) is also included in the same set, Ws e E Since the only singular point of G, in Fis 4, and fis supported by I, we infer from (4.6) that
{4.7) }.(L) c{2), VieE
Therefore, noticing that Vre EF
Lz) == Œ,G(2), ze CŠTT
we obtain from (4.7), Lemma 2.3 and Lemma 2.4, that G has no singularity on T\ {2} Since yg#0, it follows from the remarks in Section 2 that the singularity set of G is not empty, and consequently G has a single singularity at 2 Consequently, using (4.3), we conclude as in the proof of part (b) of Theorem 1.1 that (A® AD* +14 == 0 Since 440, this shows that 4 is an eigenvalue of A*, and the proof is complete
REMARK It follows from [22, p 79, Corollary 7.9] that if E is a Hilbert space and A a contraction in CE) such that limsup!'A’x! > 0, for some xe E, then
Trang 25the second conclusion of Theorem 4.1 still holds, if the hypothesis that o(4)n T is countable is replaced by the weaker hypothesis that this set is of measure zero with respect to Lebesgue measure on T
Combining the methods of this paper with the methods of {1] we obtain the following extension of [1, Theorem 1]
THEOREM 4.2 Let E be a complex Banach space and let A be an operator in (E) Assume that there exist sequences (X,)yez C E and (y;)„e„ c E* with xạ#0 and vạz#0 such that (0.1) holds Suppose that (||W„Ì)„¿„ is dominated by a Beurling sequence and that for some integer k > 0 and constant c > 0
(4.8) lJxall + l|zzll =OŒ), n —= oo
and
44.9) l|x_„ll = O(exp(cz!*)), m > 00
Then either A is a multiple of the identity operators or A has a non trivial hyper- invariant subspace
Proof First notice that (4.8) and (4.9) imply that (||x;|l)»„ez 1s dominated by a Beurling sequence, and therefore if the function G,, associated with the sequence (X Juez by (1.5), has more than one singularity, then the conclusion of the theorem follows from part (c) of Theorem 1.1 Thus in what follows we shall assume that G, has a single singularity 4, and as observed in the proof of part (a) of Theorem 1.1, it suffices to consider the case in which 2 = 1
As also noticed in previous proofs, if either 4 or A* has an eigenvalue, then the assertion of the theorem follows Thus we shall assume in the sequel that neither of these operators has an eigenvalue
We shall show that these assumptions imply that there exist non zero vectors ue and ve £* such that
(4.10) (Bu, v> =0, VBe(4),
and by Lemma 3.2, this will imply the conclusion of the theorem
To prove (4.10), we consider as in [1, Section 2] the Banach algebra B, which
fee}
consists of all analytic functions f(z) = X a„Z” in the unit disc U = {ze C: |z| < 1},
n=O œ
sụch that Š_ |z„|(1 + n)* < oo, the latter quantity serving as the norm of f in B,
nr-0
(& denotes here the integer in (4.8)) It follows from (4.8) that for every function
eo có 20
ƒ() := Š' a„z”in B,, the series W' a„x„ and Ÿ) d;y„ COnVerBe in the norms of E and
ned n=0 nd
Trang 26It is clear that the mappings
fouf), feB, and g- og), ge B,,
are bounded linear transformations from 8, into E and E* respectively By the same argument as in the proof of Lemma 3.1 we obtain that Vf, g € B,
(4.11) (Bu( f), v(g)> == (Bx, (fg)>, WBe(A)’ For every a 2 0 we denote by f, the analytic function
zed
SAz) =:(Z — l)” expa-—-
+ “
zeu,
where m =: 2k 3 As noted in [l, Section 2], the functions f, are in B,, and for every b > 0, lim f, f, in the norm of B,
aonb
We shall show that for some s > 0
(4.12) (BG,(z), t(f,)> == 0, Wze C\T, VBE (Ay First we show that (4.12) implies (4.10)
Noticing that G,(0) = x_,, and using (0.1) we obtain from (4.12), by replac- ing B by BA, that
(4.13) (Bay, t(f)ỳ =0, VBe(4
Let
x= infla > 0: <Bxạ, c(ƒ2)) =0, WBe(A)’} Since lim/,:-: ƒ, in the norm of B,, we obtain that
ate
(4.14) (Bx, vl f,)) =0, VBs(Ay
Therefore if v( f,) #0, (4.14) implies (4.10) with w=: x9 and v == o(f4)
Suppose now that v(f,) «= 0 It follows from (0.1) that v(fo) =» (A® D*¥y> and therefore by the assumption that A* has no eigenvalues, v(f))#0 Consequently, the assumption that v(f,) =: 0 implies that x > 0
A similar argument shows that u(fo>)#0, and since limu(/,) == u(fo), there
a—¬
exists a number 0 < ổ < z such that +(ƒ;)#0 Noticing that Va>0, Vb> 0, J4(Z)Ø6(2) = (2 — 1)" fore), 7EU
and that by (0.1), for every polynomial p,
Trang 27we obtain from (4.11) and the assumption that v(f,) <=: 0, that
(4.15) (Bul fy), (fa—p)> = (BA — D"xX9, o(fz)> = 0, WBe(Ay’
Since 0 <a — B <a, it follows from the definition of « that v(f,_,)4#0 There- fore (4.15) implies (4.10) with uw -= u(f,) and v = 0ƒ, _ g)
Thus it remains to prove (4.12) This is accomplished by the methods used in the proof of [1, Theorem 2], as follows
First observe that (4.8) implies that
|Gy(2)|| == Oz) — Io**, lz} a 1+
and by estimates which are similar to those used in [1, Lemma 2(a)] we obtain that (4.9) implies that
( d
I|G:()lÌ = [exp <a) lz|—> 1—
for some constant d > 0 Therefore since z =: 1 is the only singularity of G,, we deduce from [1, Lemma 3] that
(4.16) || Œa(2)J| z= O(L — zl)7#*#*#1!, Jz¡ 1+
and
b
(4.17) NG(z)|| = o[se-r—~} lz]—1—
H—ỡi
for some constant b > 0
Following [1, Section 1] we denote for every function f which is analytic in U and every weU by L,f the analytic function in U defined, for ze U\{w}, by Ly f(z) == I) =I) | As noted in [I], for Vfe B, and VweU, also L, fe B,
zZ—w
Using (3.2) and comparing Taylor coefficients we obtain VfeB,, and V Be(A)’ the identity
(4.18) (BG (z), o(f)> — f(Z)(BGYZ), ¥o> == (Bxy, (L,f)>, ze U
From this point the proof proceeds exactly as the proof of [1, Theorem 2], by replacing the estimates (16) and (17) in [1] by the estimates (4.16) and (4.17), and the identity (5) in [1] by identity (4.18) We omit the details
ReEMARK The conclusion of Theorem 4.2 also holds if the hypotheses (4.8) and (4.9) are replaced by the hypotheses
(4.19) IPnI + ly —z = O(n’), 7> CO
and
Trang 28This is proved in the same way as Theorem 4.2 by replacing the mappings f -> »(/),
fe Band f of), fe B, by the mappings f'— u'(f), fe B, and LOD, Fe Be
defined for f(z) = Š g„z”m By, by #(ƒ) = Š a,X_, and v'(f) = y đ„y_„, and
n=O n=0 n:¬0
observing that Vine N,
A™u'( fo) = (I — 4)”xạ and A**e'(fo) = UI — A*)?y Theorem 4.2 implies the following:
CoROLLARY 4.3 Let A be an invertible operator in &(E) and assume that there exist non zero vectors Xy& E and y,¢ E* such that for some integer k > Q and some constant ¢ > 0,
(4.21) 1Á xạ {DA hại] = O(n"), 1 > 00 and
(4.22) HÁT Na + APO B yy | = O(exp(cn?)), a 00 Then the conclusion of Theorem 4.1 holds for A
Proof It follows from (4.21) and (4.22) that the sequences Xz, A'%y, HEZ and y,°- Ay, ne Z satisfy the hypotheses of Theorem 4.1
A particular case of Corollary 4.3 is clearly:
CoROLLARY 4.4 ({1, Theorem 1]) Let A be an invertible operator in £(E) and assume that for some integer k > 0 and constant c > 0
A" = O(n), n— co and
"AN -= O(exp(ca¥?)), a > —oo Then the conclusion of Theorem 4.2 holds for A
Finally, we obtain from Theorem 4.2 the following extension of Theorem 1.4 and [I, Proposition 5]:
THEOREM 4.5 The conclusion of Theorem 1.4 holds if condition (1.11) in its hypotheses is replaced by the weaker condition
(4.23) lim sup “nhớ < Kexp(cn'"), neN
7¬ co m
Trang 29Proof Using Corollary 3.4 with q, = Kexp(cn'), nmeN, we obtain from (4.23) by an obvious modification of the proof of Theorem 1.4 that (0.1), (4.19) and (4.20) hold for some sequences (x,),¢z < E and (y,),ez ¢ E* with x)#0 and yq%0 Thus the conclusion of the theorem follows from the remark following the proof of Theorem 4.2
5 GENERALIZED BILATERAL WEIGHTED SHIFTS
In this section we apply Theorem 1.1 to obtain an extension of the results in [13] concerning the existence of hyperinvariant subspaces for certain bilateral weighted shifts We shall consider a larger class of operators, which we call gene- ralized bilateral weighted shifts Before defining this class, we recall some definitions Let E be a Banach space A pair of sequences (é,),cz © E and (e*), cz E”
is called a biorthogonal system, if
(5.1) (Cm 3 er» = Omn> Vin, n € z
A sequence (é,),<7 © E is called minimal if there exists a sequence (e*), <7 © E* such that {(#a)sez› (2Ÿ)„ez} forms a biorthogonal system
It follows from the Hahn-Banach Theorem that a sequence (Â,),cz â Ê is minimal if and only if for every je Z, e, is not inthe closed span of the set {e,:neZ, n#j}
A minimal sequence (e,),¢2, ¢ £ is called fundamental if its closed span coincides with E
If (eJnez < £ is a fundamental sequence, there exists a unique sequence (e*),ez © E* such that (5.1) holds We shall call (e*), <7 the dual sequence of (€,nez-
A sequence (é,)nez © E is called normalized if |le,|| = 1, Wne Z
DEFINITION Let E be a complex separable Banach space An operator A in L(E) is called a generalized bilateral weighted shift (GBWS) if there exists a fun- damental sequence (é,),¢2 ¢ £, whose dual sequence (e*), <7 is norm bounded, and a sequence of complex numbers (4,),¢z such that
(5.2) Ae, =2,€ns1, WaeZ
The sequence (/,),¢7 is called the weight sequence of A Clearly ||A,|| < ||Al], Wn e Z, hence (A,)nez € f°(Z)
RemARK According to a result of Obsepian and Pelczynski (see [18], p 44) every infinite dimensional separable Banach space contains a fundamental normalized sequence whose dual sequence is norm bounded
Trang 30some sequence of complex numbers (/,),¢2- For information on bilateral weighted shifts we refer to [13] and the references given there
Since the dual sequence of a normalized Schauder basis is norm bounded (cf [18], p 7), every BWS is also a GBWS
It is not known whether or not every BWS has a non trivial hyperinvariant subspace Partial results are proved in [13] The answer is not known even for bilateral weighted shifts in a Hilbert space, which are defined with respect to an orthonormal basis The main results known for such operators appear in [21] Remark Every GBWS clearly has a non trivial invariant subspace If A is a GBWS defined with respect to the sequence (e,),¢7, then the closed span of
the set {e,:n€Z, n > 1} is a non trivial invariant subspace for A From Theorem 1.1 we obtain the following results
THEOREM 5.1 Let A be a GBWS in Y(E) with weight sequence (A,)nez,- TỶ A#0 then each of the following two conditions implies that A has a non trivial Ayperinvariant subspace (I) (5.3) infj2„' = 0 nEZz (1D 4,40, Wave Z, and (5.4) Ỷ -+zÍ | lay A; + y; log,Â_ ;Ì )< co Lý gO
Proof Assume that A is defined with respect to the fundamental normalized sequence (@,),<7, With dual sequence (e*),-, Using (5.1) and (5.2) we obtain that
<e,, Á*e? — A„ re) ¡) =0, WneZ, VkEZ and therefore since (e,),Âz is fundamental,
(5.5) Ađe*đ = 4, 3071, WneZ
if 4; =: 0 for some je Z, then A is not injective, and therefore if A%0, ker(A) is a non trivial hyperinvariant subspace for A
Assume that 4,40, Wane Z and consider the sequence of complex numbers (4.nez defined by:
nol Rn
goed; a, JP 4, nN; o_,-= [Tp Acj, neN,
7-0 jot
Trang 31Using (5.2) and (5.5) we obtain that (0.1) holds for A and the sequences defined by (5.6)
Part I of the theorem follows from part (e) of Theorem 1.1, by observing that the sequences defined by (5.6) satisfy
lIx„~:lIIy-zll= lAnllleneaililer ll, We Z We turn now to the proof of part II
Noticing that for every complex number #0
log al| = log jal + log* a
and remembering that |le,|| = 1, Wa¢Z and that |je*|| <c, VaeZ for some constant c > 0, we obtain from (5.6) that
3 (og† ||x„|l + log? l|y„|) <
néZ 1 n 2
1 1
< log|«,{| + loge Yo —
2y 1 n? | 4 | 5 ky 1 + re
and therefore if (5.3) holds then (1.7a) is also satisfied By part (I) we may assume that (5.3) is not satisfied, and it is easily verified that this assumption implies that (1.7b) holds for some constant 6 > 0 From (5.1) we see that x, is not contained in the closed span of the set {x,:n¢Z, 40} and that yo is not contained in the closed span of the set {y,:7¢€Z, n40} Thus the desired conclusion follows from part (d) of Theorem 1.1
COROLLARY 5.2 Let A be a GBWS with weight sequence (A,),¢7- If AX0 and
5.7 [1 = lll
( ) 2 1 + In < Os
then A has a non trivial hyperinvariant subspace
Proof If (5.3) holds the assertion follows from part I of Theorem 5.1 Thus we may assume that inf|/,| > 0 Remembering that (/,),¢z €4°(Z), we obtain that
nez ,
there exists a constant d > 1 such that
sup
4ˆ! <|Àj|<4 VneZ
Since
eet lar < is) <a < 0, 1 — jx| |
Trang 32
we deduce from (5.7) that
oo Yo (IIogi2,I| + ogi2_„Íl) < eo 1 Pow Gg
~~ H
Ì
oo 1
Vị :(lD
i.e
` ioe rail + losls ) Xe | ns:0 jon ATP
and therefore > log|À_, y lost] +] MN
<5 ¥ —-!— (Hog all + Hoga) < 00, n=0 l+n
and the assertion follows from part II of Theorem 5.1
Theorem 5.1 extends the result of [13, Theorem 4], where the existence of non trivial hyperinvariant subspaces is proved for bilateral weighted shifts which satisfy certain conditions which are more restrictive than (5.3)
For bilateral weighted shifts which are defined with respect to a rotation inva- riant basis (see definition in |13}, p 776), Corollary 5.2 coincides with [13, Corollary 2] REMARKS 1] If |2,| > 1, We Z, then the proof of Corollary 5.2 shows that conditions (5.3) and (5.7) are equivalent
2 Condition (5.7) in Corollary 5.2 can be replaced by the more general condition, that for some constant a 2 0
lant — a
(*) 4! < co
nez 1 +H!
Indeed for a == 0, condition (*) implies (5.3) and the assertion follows from part I of Theorem 5.1 If a > 0, the assertion follows from Corollary 5.2 by observing that (2~12,)„ez is the weight sequence of operator a7 1A
3 If A is a GBWS which is invertible, we obtain from (5.2) that [Ani > HAT! VneZ
Thus a GBWS (in particular a BWS) which satisfies (5.3) is not invertible If A is a
BWS on a Hilbert space, which is defined with respect to an orthonormal basis,
Trang 334 If A is a BWS on a Banach space which is defined with respect to an uncon- ditional normalized basis (see definition in [18], p 15), then it follows from [18, Proposition 1.c.7] that (5.3) holds if and only if A is not invertible However as we shall show in Proposition 6.2 this is not true in general even for a BWS which is defined with respect to a rotation invariant basis This will disprove the conjec- ture in [12, p 543] (See also the remark following Proposition 6.2.)
5 It is claimed in [13, p 772 and p 777] that if A is a non invertible BWS which is defined with respect to a rotation invariant basis, then every element in (A)’ is the limit, in the strong operator topology, of a sequence of polynomials in A But the proof of this claim is not correct, since it is based on the assertion that it is proved in [12], that every element in (A)’ can be identified with a power series (in the sense described in [12]) But this fact is proved in [12, Theorem 4(2)} only for bilateral weighted shifts for which (5.3) holds, and as mentioned in Remark 4, there exist bilateral weighted shifts (even defined with respect to a rotation inva- riant basis) which are not invertible, but do not satisfy (5.3)
However, as Professor Herrero has shown us, the gap in the proof can be easily corrected, so that the result remains true also if (5.3) is not satisfied Conse- quently, the assertion in the proof of [13, Theorem 4], that every non invertible BWS which is defined with respect to a rotation invariant basis has a non trivial hyperinvariant subspace, is correct also if (5.3) does not hold
6 EXAMPLES, COMMENTS AND PROBLEMS
In this section we give some examples, make some comments and pose some problems
We begin with an example of an operator which satisfies the hypotheses of part (a) of Theorem 1.1 and those of Theorem 1.2 but does not satisfy the hypo- theses of Theorems 1.3 and 1.4
Let (11,)9? be the sequence of integers defined by:
n; =j + 2), even, and n, = (j + 1)®, #odd, and consider the sequence (/,),¢z defined by:
"
A,= 2° for nj <a < anja, 7 =9,1, , and
A, == Ar} for n < 0
Let £ be a complex Hilbert space with orthonormal basis (e€,),¢z, and let 4 be the invertible BWS defined on E£ by
Trang 34Using [21, p 59 and p 67] it is easy to see that
(6.1) jA4l=2”, VneZ
and therefore A does not satisfy the hypotheses of Theorems l.3 and 1.4
It also follows from {2!1, Theorem 5] and (6.1) that the spectrum of A con- sists of the anulus {ze(:2”' < !z < 2}
A simple computation shows that
[Avegii <1, Wane Z and 'A*"ei <1, VneZ
and therefore A satisfies the hypotheses of part (a) of Theorem 1.1 and Theorem 1.2 Next we consider generalized bilateral weighted shifts which are defined on some spaces of functions on T
In what follows B will denote a homogeneous Banach space on T, in the sense of [17, p 14], and e, will denote for every x € Z, the function on T defined by e,(w) -= wv", we T
We shali also assume that e,¢ B, Wane Z and that, je„|g:-:l, Vne€Z, and also that V fe B and Vue Z, e, fe B and jie fp = lifics-
Since fils 2 flr» f¢B, the sequence of linear functionals ef, ne Z defined on B by
C/.e‡>= Âu), feb
are in B*, and |ie?tp = 1, Ware Z
It is clear that {(e,),ez>(€e)nez} is a biorthogonal system, and according to {17, p 15, Theorem 2.12], (,),¢z is a fundamental sequence in B
Examples of spaces which satisfy all the above conditions are the spaces C(T) and L°(T) for 1 < p < co The sequence (e,),¢z is a Schauder basis in L°(T) for 1 < p < oo but not in LT) and C(T) (see [17], Chapter II)
In the rest of this section the term generalized bilateral weighted shift on B will mean a GBWS which is defined with respect to the sequence (e,),¢z-
It follows easily from the hypotheses on B, that the generalized bilateral weighted shifts on B are the operators of the form V-P, where V is the isometry defined on B by
(6.2) Vf=e-f, feB
and P is a Fourier multiplier on B, that is an operator in Y(B) such that for some sequence of complex numbers (p,), <2
Trang 35(p,)nez-Let M(T) denote the space of all (complex) Borel measures on T It is known (see [17], p 39) that every measure » ¢ M(T) defines a Fourier multiplier P on B by
(6.3) Pf=p«f, feB
where y xf denotes the convolution of j with f The sequence (p,),,<7, which corres- ponds to this multiplier is given by p, = fi(n), ne Z, where fi(n) denotes the n-th Fourier-Stieltjes coefficient of y
From these remarks we see that every measure » € M(T) defines on Ba GBWS 4 which is given by
(6.4) Af=e(uxf), fe B
The weight sequence of this shift is clearly ((n)),,¢z -
An immediate consequence of Theorem 5.1 and Corollary 5.2 is:
THEOREM 6.1 Let be a measure in M(T), 140, and consider the operator A defined on B by (6.4) Each of the following two conditions implies that A has a non trivial hyperinvariant subspace:
nezZ, 1 + \n|
(6.6) inf |fi(n)| = 0
„c2
It is known that every Fourier multiplier on the spaces [4(T) and C(T) is of the form (6.3) for some measure y € M(T) (cf [10], Section 16.32) Consequently the géneralized bilateral weighted shifts on these spaces are exactly the operators of the form (6.4)
It is well known and easily verified, that the Fourier multipliers on L°(T) can be identified (in the obvious way) with £°(Z) (see [10, Section 16.1.2(4)}) This corresponds to the simple known fact that a sequence of complex numbers (4,),¢z is the weight sequence of a BWS on a Hilbert space, which is defined with respect to an orthonormal basis, if and only if (A,),¢z € £°(Z) On the other hand no charac- terization is known for the multipliers on the spaces L?(T) for 1 < p < co, p#2 Partial results can be found in [10, Section 16.4]
R Gellar conjectured in [12, p 543] that if A is a BWS on a Banach space, with weight sequence (A,),¢7 and
m+n Yn m+n 1n
Ñ, = lim [sp il vl R, = lim (int Ul vl mòQœ (G2 7m3 1 nocœ Ẳm€Z j=m+1
Trang 36Noticing that R, = 0 if and only if inf j/,! = 0, we see that the truth of the
zEZ
conjecture would imply that if inf j4,, > 0 then A is invertible Thus the conjecture
néZ
is disproved by the following:
PROPOSITION 6.2 Jf 1 < p < 2, there exists a BWS A on L(T) whose weight sequence (2,)nez, consists of real numbers such that 1, 21, WneZ, but A is not invertible
Proof Let 1 < p <2 be fixed According to a theorem of Igari[16, Theorem 6], there exists a measure 4 € M(T) whose Fourier-Stieltjes coefficients are real and satisfy fi(a) > 1, such that the Fourier multiplier defined on L’(T) by (6.3) is not invertible Consider the BWS A on L?(T) defined by (6.9) with this measure Since the operator V (defined by (6.2)) is invertible, it follows that A is not invertible, and the Proposition is proved
REMARKS 1) After a first draft of the paper was circulated, Professor Herrero informed us that Gellar’s conjecture was also disproved by R Gellar and R Silber in Proc Amer Math Soc., 61(1976), 225—226 Their example relies on a Banach Space introduced by Nakano, and is somewhat less ‘‘natural’’ than the example in Proposition 6.2
2) The basis (e,),¢z in L’(T) for 1 < p <2, is clearly rotation invariant accord- ing to the definition in [13, p 776] This is also the case in the example of Gellar and Silber
It is natural to consider the more general class of operators on the spaces L(T), 1 < p < 00, which is obtained by replacing the function e, in (6.4) by an arbi- trary L%(T) function That is, for every function g ¢ L™(T), and every measure i € M(T), one can consider the operator A on L?(T), ! < p < co, deñned by
(6.7) Af= O(usf), fe LT)
For these operators the existence of non trivial invariant subspaces is not known in general even when yp is the unit point mass concentrated at some we T In this
case the corresponding operators are given by
(6.8) Af = @-fy, fe LT)
where f,(z) = f(w7'z), ze T
Trang 37It would be interesting to find conditions on g and yp, in addition to those given by Davie in [7] and by Theorem 6.1 of the present paper, which imply that the operators defined by (6.7) possess non trivial invariant (or hyperinvariant) subspaces One such condition is that yw is absolutely continuous, or for p = 2, more generally, that lim ju(n) = 0 It is easy to show that, in this case, the operator
Rat 00
given by (6.7) is compact for every o  Lđ(T) Therefore by the well know result of Lomonosov (cf [20], p 158), these operators have non trivial hyperinvariant subspaces (if 140 and ø #0)
We conclude with the following:
PROBLEM Lei @ = e, + €_, Does the operator, defined by (6.8) with this function g, have a non trivial invariant subspace for every we T?
This problem is of some interest in view of the fact that one can show that if w is not a rooth of unity, then the two operators given by (6.8) with @ = e_, and =, respectively, do not possess a common non trivial invariant subspace
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AHARON ATZMON Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, U.S.A Permanent address: Technion — I.1.T., Haifa,
Israel