OPERATORS WITH COMMON HYPERCYCLIC SUBSPACES

Transcription

1 OPERATORS WITH COMMON HYPERCYCLIC SUBSPACES R. ARON, J. BÈS, F. LEÓN AND A. PERIS Abstract. We provide a reasonabe sufficient condition for a famiy of operators to have a common hypercycic subspace. We aso extend a resut of the third author and A. Montes [22], thereby obtaining a common hypercycic subspace for certain countabe famiies of compact perturbations of operators of norm no arger than one. 1. Introduction It is known that for any separabe infinite dimensiona Banach space X, there is a continuous inear operator T : X X which is hypercycic; that is, there is a vector x such that the set {x, T x,..., T n x,...} is norm dense in X ([2], [5]). Moreover, a simpe Baire category argument shows that the set HC(T ) of such so-caed hypercycic vectors x is a dense G δ in X [21], and its inear structure is we understood: Whie HC(T ) must aways contain a dense subspace ([9], [20]), it not aways contains a cosed infinite dimensiona one; see [16] for a compete characterization of when this occurs. (Throughout, when we say that HC(T ) contains a vector space V we mean of course that every x V except x = 0 is hypercycic for T.) Date: September 6, Thus, for exampe it was shown 1991 Mathematics Subject Cassification. Primary: 47A16. Key words and phrases. Hypercycic vectors, subspaces, and operators; universa famiies. 1

2 2 R. ARON, J. BÈS, F. LEÓN AND A. PERIS that for the simpest exampe of a hypercycic operator on a Banach space, namey the Roewicz operator B 2 : 2 2, B 2 (x 1, x 2, ) = 2(x 2, x 3, ), HC(B 2 ) contains an infinite dimensiona vector space but that this vector space cannot be cosed [25, Theorem 3.4]. In recent years, an increasing amount of attention has been paid to the set T F HC(T ) of common hypercycic vectors of a given famiy F of hypercycic operators acting on the same Banach space X. A trivia extension of the Baire argument auded to above tes us that T F HC(T ) is a dense subset of X whenever F is countabe. Moreover, L. Berna and C. Moreno [7] showed this set contains a dense vector space if we ask in addition that the members be hereditariy hypercycic. Finay S. Grivaux proved that this additiona hypothesis can be suppressed [17, Proposition 4.3]. Other important recent work is by E. Abakumov and J. Gordon [1], who showed that the uncountabe intersection {λ C λ >1} HC(B λ ), where B λ is the Roewicz operator with 2 repaced by λ. In fact it is simpe to derive from this that the intersection contains a dense subspace of 2. On the other hand, in [4] F. Bayart shows that under the assumption of a strong form of the hypercycicity condition, uncountabe coections of hypercycic operators can indeed contain an infinite dimensiona cosed subspace of common hypercycic vectors. Simiar resuts were obtained by G. Costakis and M. Sambarino [13], who aso provided a criterion for the existence of common hypercycic vectors. Our interest here wi be in the foowing probem:

3 OPERATORS WITH COMMON HYPERCYCLIC SUBSPACES 3 Probem 1. Let F be a countabe famiy of operators acting on a Banach space X. When does T F HC(T ) contain a cosed infinite dimensiona subspace? After giving a natura exampe that indicates that the Abakumov- Gordon, Bayart and Costakis-Sambarino situations fai to hod in genera for common subspaces, we prove the main resut of this note, which extends a resut by A. Montes [25, Theorem 2.1] by providing a reasonabe sufficient condition on a countabe famiy of hypercycic operators acting on a Banach space to have a common infinite dimensiona hypercycic subspace (Coroary 3.5). We then appy this to extend a resut of the third author and A. Montes [22], thereby obtaining a common hypercycic subspace for certain countabe famiies of operators of the form T = U + K where U 1 and K is compact. 2. exampe Exampe 2.1. Let X = H be a separabe, infinite-dimensiona Hibert space, and et S H be the unit sphere of H. Let (w n ) be a sequence of positive scaars satisfying im n inf k ( n w k+j ) 1 n 1 and im sup n w j =. For each h in S H, et {e(h) n : n 1} be a basis of H with e(h) 1 = h, and et T h : H H be the corresponding uniatera weighted backward shift defined by 0 if n = 1 (1) T h e(h) n = w n e(h) n 1 if n 2,

4 4 R. ARON, J. BÈS, F. LEÓN AND A. PERIS So T h has a hypercycic subspace [23, Coroary 2.3]. Aso, notice that F = {B h : h S H } satisfies that for a 0 y in H, T y y y = 0. That is, F is a famiy of operators, each one having a hypercycic subspace, but such that there is no hypercycic vector common to a members of F. Let us aso observe that in [1] the authors mention that there is no common hypercycic vector for the famiy of hypercycic operators {λb δb : λ, δ > 1}. It is easy to see that no operator in this famiy admits a hypercycic subspace. 3. A sufficient condition for a common hypercycic subspace We prove the main resut in the more genera setting of universaity. Given a sequence F = {T j } j N of bounded operators acting on a Banach space X, we say that a vector x X is universa for F if { T x : T F } is dense in X; the set of such universa vectors is denoted HC(F). The sequence F is said to be universa (respectivey, densey universa ) provided HC(F) is non-empty (respectivey, dense in X). F is caed hereditariy universa (respectivey, hereditariy densey universa) provided {T nk } k N is universa (respectivey, densey universa) for each increasing sequence (n k ) of positive integers. For more on the notion of universaity, see [15] and [19]. A resut simiar to the foowing theorem is proved in [10] for a (unique) sequence of universa operators in the context of Fréchet spaces.

5 OPERATORS WITH COMMON HYPERCYCLIC SUBSPACES 5 Theorem 3.1. Let T n,j (n, j N) be bounded operators on a Banach space X, and et Y be a cosed subspace of X of infinite dimension. Suppose that for each n N i) {T n,j } j N is hereditariy densey universa, and ii) im j T n,j x = 0 for each x in Y. Then there exists a cosed, infinite dimensiona subspace X 1 of X so that {T n,j x} j N is dense in X for each non-zero x X 1 and n N. In particuar, X 1 is a universa subspace of {T n,j } j N for each n N. Lemma 3.2. Let T n,j (n, j N) be bounded operators on a Banach space X so that for each fixed integer n the famiy {T n,j } j 1 is densey universa. Then the set n=1hc({t n,j } j 1 ) of common universa vectors to every sequence {T n,j } j N is dense in X. Proof. n=1hc({t n,j } j 1 ) is a countabe intersection of dense G δ subsets of the Baire space X [18, Satz 1.2.2]. Proof of Theorem 3.1. Reducing the subspace Y if necessary, we may assume it has a normaized Schauder basis (e j ) j. Let (e j) be its associated sequence in Y of coordinate functionas, that is, so that e j(e i ) = δ i,j for i, j N. Let A(Y, X) denote the norm cosure of the subspace { n } x j e j( ) : n N, x 1,..., x n X. For each T in B(X), define L T : A(Y, X) A(Y, X) by L T V := T V. We make use of the foowing emma, whose proof foows that of Theorem 3.1. Anaogous versions of this emma are proved in [10] for severa operator ideas (nucear, compact, approximabe), in a more genera context, by using tensor product techniques deveoped in [24].

6 6 R. ARON, J. BÈS, F. LEÓN AND A. PERIS Lemma 3.3. Suppose {T j } j N is a sequence of bounded operators on X that is hereditariy densey universa. Then {L Trj } j 1 is a hereditariy densey universa sequence of operators on A(Y, X), for some increasing sequence (r j ) of positive integers. Now, notice that by (i) and Lemma 3.3, for each fixed n N there exists a sequence of positive integers (r n,j ) j so that the sequence of operators {L Tn,rn,j } j N is hereditariy densey universa on the Banach space A(Y, X). By Lemma 3.2, there exists V in A(Y, X) that is universa for every sequence {L Tn,rn,j } j N, and hence universa for every {L Tn,j } j N, too (n N). Mutipying V by a non-zero scaar if necessary, we may assume that V < 1 2. Consider now X 1 := (i + V )(Y ), where i : Y X is the incusion. For each x Y, (i + V )x x V x 1 2 x. So i + V is bounded beow and X 1 is cosed and of infinite dimension. Notice that {T n,j V x} j N is dense in X for every 0 x Y and every n N. Indeed, given ɛ > 0, et z X be arbitrary, and et S be a finite rank operator in A(Y, X) such that Sx = z. By Lemma 3.3, for each n there is some T n,j such that T n,j V S < ɛ. In particuar, x T n,j V x Sx = T n,j V x z < ɛ. The theorem now foows from condition (ii). Proof of Lemma 3.3. Since {T j } j N is hereditariy densey universa on X, it foows from [6, Theorem 2.2] that there exists a dense subspace X 0 of X, an increasing sequence of positive integers (r j ) and (possiby discontinuous) inear mappings S j : X 0 X (j N) so that (2) T rj, S j, and (T rj S j I) j 0 pointwise on X 0. Now, consider A 0 := {V A(Y, X) : V (Y ) X 0 and dim(v (Y )) < }.

7 OPERATORS WITH COMMON HYPERCYCLIC SUBSPACES 7 Then A 0 is dense in A(Y, X), and it foows from (2) that L Trj, L Sj, and [L Trj L Sj I] j 0 pointwise on A 0. So {L Trj } j 1 is hereditariy densey universa on A(Y, X), by [6, Theorem 2.2]. Remark 3.4. An aternative constructive proof of Theorem 3.1 may be done with the arguments from [25, Theorem 2.2]. The proof here is much simper, and foows arguments from [10] and [11]. Coroary 3.5. Let T ( N) be operators acting on a Banach space X. Suppose there exists a cosed, infinite dimensiona subspace Y of X, increasing sequences (n,q ) q of positive integers, and scaars c,q so that for N i) {c,q T n(,q) } q N is hereditariy universa, and ii) im q c,q T n(,q) x = 0 for each x in Y. Then there exists a cosed, infinite dimensiona subspace X 1 of X so that {c,q T n(,q) x} q N is dense in X for each non-zero x X 1 and each N. That is, X 1 is a supercycic subspace for T for every N. Moreover X 1 is a hypercycic subspace for T for every N if the constants c,q are of moduus one. 4. An Appication to Countabe Famiies of Operators We now appy Theorem 3.1 to show the foowing extension of [22, Theorem 4.1] to countabe famiies of operators. Theorem 4.1. Let F = {T = U +K : N} be a famiy of operators acting on a common Banach space X. Suppose that for each N a) U 1, K is compact, and

8 8 R. ARON, J. BÈS, F. LEÓN AND A. PERIS b) {T n,q } q 1 is hereditariy universa, for some increasing sequence (n,q ) q 1 of positive integers. Then the operators in F have a common hypercycic subspace. To show Theorem 4.1, we make use of the foowing emmas. The first one foows from a sight modification of a proof by Mazur [14, p 38-39]. The second one is [16, Lemma 2.3] The ast one is proved at the end of this section. Lemma 4.2. Let (X n ) be a sequence of cosed, finite-codimensiona subspaces of X, with X n X n+1 (n 1). Then there exists a normaized basic sequence (e n ) so that e n beongs to X n for a n 1. Lemma 4.3. [16, Lemma 2.3] Let {T n,q } q be hereditariy hypercycic ( N). Then there exists a dense subset X 0 of X and, for each N, a subsequence (r,q ) q of (n,q ) q so that im q T r,q x = 0 (x X 0 ). Lemma 4.4. Let X and Z be Banach spaces, and et K,n : X Z be compact operators (, n 1). Given ɛ > 0, there exist cosed inear subspaces X n of finite codimension in X (n 1) so that i) X n X n+1 ii) K,n x ɛ x (x X n, 1 n) Proof of Theorem 4.1. Because {T n,q } q 1 is hereditariy densey universa for each N, by Theorem 3.1 it suffices to get a cosed, infinite dimensiona subspace Y of X and subsequences (m,q ) q of (n,q ) q so that im q T m,q x = 0 (x Y, N).

9 OPERATORS WITH COMMON HYPERCYCLIC SUBSPACES 9 For each pair of positive integers n and, et K,n be the compact operators defined by T n = (U + K ) n = U n + K,n. Appy Lemma 4.4 to get cosed, finite codimensiona subspaces X n of X satisfying a) X n X n+1 (3) b) K,n x x (x X n, 1 n). By Lemma 4.2, we can pick a normaized basic sequence (e n ) in X so that e n X n (n N). Let K > 0 be the basis constant of (e n ), and pick a decreasing sequence of positive scaars, (ɛ m ), so that n=1 ɛ n < 1 2K. By Lemma 4.3, there exist subsequences (ñ,q ) q of (n,q ) q and a dense subspace X 0 of X so that (4) im q T ñ,q x = 0 (x X 0 ). Pick a sequence (z m ) in X 0 so that (5) e n z n < Notice that e n z n < ɛ n max{ T i ɛ n :, i n. }. (n 1) and, because (e n ) is normaized, e n(x) 2K x (n 1) for a x in Y 0 = span{e 1, e 2,... }, where (e n) is the sequence of functiona coefficients associated with the Schauder basis (e n ) of Y 0. Hence n=1 e n e n z n < 2K n=1 ɛ n < 1, and so any subsequence (z nk ) of (z m ) is equivaent to the corresponding basic sequence (e nk ) [14, p 46]. We et Y := span{z nk : k 1}, where (z nk ) (z n ) is defined as foows. Let n 0 := 1. For N, choose m,1 in (ñ,q ) so that T m,1 z n0 < ɛn 0 2. Aso, et n 1 := m 1,1. Next, for each N, since z n0, z n1 X 0, we may appy (4) to get m,2 (ñ,q ) q which satisfies the foowing conditions. m,2 > max{2, n 1, m,1 } T m,2 z ni < ɛn i 2 2 i = 0, 1.

10 10 R. ARON, J. BÈS, F. LEÓN AND A. PERIS Aso, et n 2 := max 1 2 {m,2 }. Continuing this process we get, for each N, an integer m,s in (ñ,q ) q so that i) m,s > max{s, n s 1, m,s 1 } (6) ii) T m,s z ni < ɛn i i = 0,..., s 1, 2 s where n r = max 1 r {m,r } for each r N. It suffices to show that T m,s 0 pointwise on Y ( N). Let 0 z = s α jz nj in Y, N be fixed, and s be arbitrary. Then (7) T m,s z = s 1 α j T m,s z nj + Notice that α j 2L z (z nk ). By (6.ii), s 1 (8) α j T m,s z nj < α j T m,s (z nj e nj )+T m,s ( α j e nj ). (1 j), where L is the basis constant of s 1 α j ɛ n j 2 L z s 1 s 2 s 1 Aso, by (6i) and (5) (9) α j T m,s (z nj e nj ) 2L z ɛ nj. Finay, since X ns X m,s (10) T m,s and U 1, by (3b) α j e nj = (U m,s + K,m,s )( 2 α j e nj ɛ nj. α j e nj ) (s ). So by (7), (8), (9), and (10), im s T m,s z = 0. We finish the proof of Theorem 4.1 by showing Lemma 4.4. Proof of Lemma 4.4. Let n 1 and ɛ > 0 be fixed. K,n : Z X is compact, there exist z,n,1,..., z,n,k,n Because each in X so that (11) K,n(B Z ) k,n i=1 B(z,n,i, ɛ).

11 OPERATORS WITH COMMON HYPERCYCLIC SUBSPACES 11 For each positive integer s, et X s := s n=1 n =1 k,n i=1 Ker(z,n,i ). So each X s is cosed and of finite codimension in X, and X s X s+1 (s 1). Now, et x X n, and et 1 n be fixed. By the Hahn-Banach theorem, there is a functiona x of norm one so that K,n x = K,n x, x. By (11), we may choose 1 j k,n so that K,n x z,n,j < ɛ. Hence, because x is in X n Ker(z,n,j ), K,nx = x, K,n x z,n,j ɛ x. The proof of Theorem 4.1 is now compete. We d ike to finish with the foowing two probems. Probem 2. Let T 1, T 2 be two operators acting on a Banach space X. Suppose each of T 1, T 2 has a hypercycic subspace (i.e., a cosed, infinite dimensiona subspace of hypercycic vectors). Must they share a common hypercycic susbpace? Probem 3. Let T 1, T 2 be two hereditariy hypercycic operators acting on a Banach space X, with a common hypercycic subspace. Must there exist sequences (n,q ) q ( = 1, 2) and a cosed infinite dimensiona subspace Y of X so that {T n,q } q is hereditariy hypercycic and T n,q 0 q pointwise on Y ( = 1, 2)? References [1] E. Abakumov and J. Gordon, Common hypercycic vectors for mutipes of the backward shift, J. Funct. Ana. 200 (2003), [2] S. I. Ansari, Existence of hypercycic operators on topoogica vector spaces, J. Funct. Ana. 148 (1997), [3] S. I. Ansari, Hypercycic and cycic vectors, J. Funct. Ana. 128 (1995), [4] F. Bayart, Common hypercycic subspaces, preprint. [5] L. Berna-Gonzáez, On hypercycic operators on Banach space, Proc. Amer. Math. Soc. 127 (1999),

12 12 R. ARON, J. BÈS, F. LEÓN AND A. PERIS [6] L. Berna-Gonzáez and K.-G. Grosse Erdmann, The Hypercycicity Criterion for sequences of operators, Studia Mathematica, 157 (2003), [7] L. Berna-Gonzáez and M. C. Caderón-Moreno, Dense inear manifods of monsters, J. Approx. Theory 119 (2002), [8] J. Bès and A. Peris, Hereditariy Hypercycic Operators, J. Funct. Ana., 167 (1999), [9] P. Bourdon, Invariant Manifods of Hypercycic Vectors, Proc. Amer. Math. Soc. 118 No. 3 (1993), [10] J. Bonet, F. Martínez-Giménez, and A. Peris, Universa and chaotic mutipiers on operator ideas, preprint. [11] K. C. Chan and R. D. Tayor, Hypercycic subspaces of a Banach space, Integr. equ. oper. theory, 41 (2001) [12] J. Conway, A course in Functiona Anaysis, 2nd edition, Springer-Verag, New York, [13] G. Costakis & M. Sambarino, Genericity of wid hoomorphic functions and common hypercycic vectors, Adv. Math., to appear. [14] J. Dieste, Sequences and Series in Banach Spaces, Springer-Verag, New York, 1984 [15] G. Godefroy and J. H. Shapiro, Operators with dense, invariant cycic vector manifods, J. Funct. Ana. 98, (1991). [16] M. Gonzáez, F. León-Saavedra, and A. Montes-Rodríguez, Semi-Fredhom Theory: hypercycic and supercycic subspaces, Proc. London Math. Soc.(3) 81 (2000), n -1, [17] S. Grivaux, Construction of operators with prescribed behaviour, Arch. Math., to appear. [18] K.-G. Grosse-Erdmann, Hoomorphe Monster und universee funktionen, Mitt. Math. Sem. Giessen 176 (1987). [19] K.-G. Grosse-Erdmann, Universa Famiies and Hypercycic Operators, Bu. Amer. Math. Soc. 36 (1999), [20] D. Herrero, Limits of hypercycic and supercycic operators, J. Funct. Ana. 99 (1991) [21] C. Kitai, Invariant Cosed Sets for Linear Operators, Ph. D. Thesis, Univ. of Toronto, 1982.

13 OPERATORS WITH COMMON HYPERCYCLIC SUBSPACES 13 [22] F. León-Saavedra & A. Montes-Rodríguez, Linear structure of hypercycic vectors, J. Funct. Ana. 148 (1997), [23] F. León-Saavedra & A. Montes-Rodríguez, Spectra Theory and Hypercycic Subspaces Trans. Amer. Math. Soc. 353 (2001), n -1, [24] F. Martínez-Giménez & A. Peris, Universaity and chaos for tensor products of operators, J. Approx. Theory, to appear. [25] A. Montes-Rodríguez, Banach spaces of hypercycic vectors, Michigan Math. J. 43 (1996), Department of Mathematics Kent State University Kent, Ohio 44242, USA aron@mcs.kent.edu Department of Mathematics and Statistics Bowing Green State University Bowing Green, OH 43403, USA jbes@math.bgsu.edu Escuea Superior de Ingeniería Universidad de Cadiz c/sacramento, Cadiz, SPAIN feon@uca.es E.T.S. Arquitectura D. Matemàtica Apicada Universitat Poitècnica de Vaència E Vaència, SPAIN aperis@mat.upv.es