Bounded point evaluations for cyclic Hilbert space operators

@ Applied General Topology c Universidad Polit´ecnica de Valencia Volume 4, No. 2, 2003 pp. 301–316 Bounded point evaluations for cyclic Hilbert space operators A. Bourhim∗ Dedicated to Professor S. Naimpally on the occasion of his 70th birthday. Abstract. In this talk, to be given at a conference at Seconda Universit`a degli Studi di Napoli in September 2001, we shall describe the set of analytic bounded point evaluations for an arbitrary cyclic bounded linear operator Ton a Hilbert space Hand shall answer some questions due to L. R. Williams. 2000 AMS Classification: Primary 47A10; Secondary 47B20. Keywords: cyclic operator, bounded point evaluation, single-valued extension property, Bishop’s property (β). 1. Introduction. Throughout this paper, L(H) will denote the algebra of all linear bounded operators on an infinite–dimensional separable complex Hilbert space H. Let T∈ L(H) be a cyclic operator on Hwith cyclic vector x∈ H i.e., the linear subspace {p(T)x:ppolynomial}is dense in H. A complex number λ∈Cis said to be a bounded point evaluation for Tif there is a positive constant M such that for every polynomial p, |p(λ)| ≤ Mkp(T)xk; equivalently, if λinduces a continuous linear functional on Hwhich maps p(T)x to p(λ) for every polynomial p. Therefore, it follows from the Riesz Representation Theorem that a complex number λ∈Cis a bounded point evaluation for Tif and only if there is a unique vector k(λ)∈ H such that (1.1) p(λ) = hp(T)x, k(λ)ifor every polynomial p. The set of all bounded point evaluations for Twill be denoted by B(T). A point λ∈B(T) is called an analytic bounded point evaluation for Tif there is ∗This research is supported in part by the Abdus Salam ICTP, Trieste, Italy. 302 A. Bourhim an open neighborhood Oof λcontained in B(T) such that for every y∈ H, the complex function bydefined on B(T) by by(λ) := hy, k(λ)i, is analytic on O. The set of all analytic bounded point evaluations for Twill be denoted by Ba(T). An operator T∈ L(H) is said to be subnormal if it has a normal extension, i.e., if there is a normal operator Non a Hilbert space K, containing H, such that His a closed invariant subspace of Nand the restriction N|H coincides with T. The operator Tis said to be hyponormal if kT∗xk≤kTxkfor every x∈ H, where T∗denotes the adjoint of T. Note that every subnormal operator is hyponormal, with the converse false (see [8] and also Example 5.6). Using Bran’s theorem [5] and the maximum modulus principle for analytic functions, Tavan T. Trent proved in [20] that for every cyclic subnormal operator T∈ L(H), we have Ba(T) = Γ(T)\σap(T). Here, σap(T) denotes the approximate point spectrum of T, that is the set of complex numbers λfor which there is a sequence (xn)nof elements of the unit sphere of Hsuch that lim n→+∞k(T−λ)xnk= 0, and Γ(T) denotes the compression spectrum of T, that is the set of complex numbers λsuch that the range of (T−λ) is not dense in H. More informations about bounded point evaluations for cyclic subnormal operators can be found in [8]. In [21], L. R. Williams followed Trent’s method to shown that (1.2) Γ(T)\σap(T)⊂Ba(T) for every cyclic operator T∈ L(H) and posed the following question. Question 1.1. Let T∈ L(H)be a cyclic operator. Is Ba(T) = Γ(T)\σap(T)? Note that, in general, the basic spectral properties of subnormal operators remain valid for hyponormal operators. Thus we pose the following weaker question. Question 1.2. Is Ba(T) = Γ(T)\σap(T)for every cyclic hyponormal operator T∈ L(H)? In this paper, we shall explain more about bounded point evaluations for cyclic Hilbert space operators from the point of view of local spectral theory and shall answer the above questions. In Section 2, we give a complete description of the set of analytic bounded point evaluations for arbitrary cyclic operators and derive some consequences from it. In Section 3, we give a necessary and sufficient condition for unilateral weighted shift operators to satisfy Trent’s result, and exhibit some operators which provide a negative answer to Question 1.1. In Section 4, we generalize a result of L. Yang which allows us to give a positive answer to Question 1.2. As a corollary, we get that two quasisimilar cyclic hyponormal operators have equal approximate point spectra; this result is a generalization of Theorem 4 of [16]. In Section 5, we show that if T∈ L(H) is a cyclic operator for which the span of the eigenvectors of T∗associated with Bounded point evaluations for . . . 303 a connected component of Ba(T) is dense in H, then Tis without eigenvalues and has a connected spectrum. Some related examples are given. Before going further, we need to introduce some notations and recall some basic notions concerning local spectral theory; we refer to the monographs [7] and [13] for further informations. For an operator T∈ L(H), we denote as usual by σ(T) := {λ∈C:T−λis not invertible},ρ(T) := C\σ(T), σp(T) := {λ∈C:T−λis not injective}, ker T, and ranTthe spectrum, the resolvent set, the point spectrum, the kernel, and the range of T, respectively. For a subset Mof H, we use cl(M), and WMto denote the closure of M, and the closed linear subspace generated by M, respectively. For a subset Fof C, we use F, and Fr(F) to denote the complex conjugates of the points in F, and the boundary of F, respectively. For an open subset Uof C, we let O(U, H) denote the space of analytic H−valued functions on U. It is a Fr´echet space when equipped with the topology of uniform convergence on compact subsets of U and the space Hmay be viewed as simply the constants in O(U, H). One says that an operator T∈ L(H) possesses Bishop’s property (β) if for each open subset Uof C, the multiplication operator TU:O(U, H)−→ O(U, H), f 7−→ (T−z)f is injective with closed range. M. Putinar [14] has shown that hyponormal operators have Bishop’s property (β). Recall that Tis said to have the single– valued extension property provided that, for every open subset Uof C, the only analytic H−valued solution fof the equation (T−λ)f(λ) = 0,(λ∈U), is the identically zero function f≡0 on U. Equivalently, if for every open subset U of C, the mapping TUis injective. A localized version of this property dates back to J. K. Finch [10] and can be defined as follows (see [1], and [10]). An operator T∈ L(H) is said to have the single–valued extension property at a point λ∈Cif for every open disk Ucentered at λ, the mapping TUis injective. Let <(T) := λ∈C:Tdoes not have the single-valued extension property at λ. Obviously, <(T) is an open subset contained in σp(T) and is empty precisely when Thas the single–valued extension property. The local resolvent set of T at a vector y∈ H, denoted by ρT(y), is the union of all open subsets Uof C for which y∈ranTU. The local spectrum of Tat yis σT(y) := C\ρT(y); it is a closed subset of σ(T). In the sequel, T∈ L(H) will be a cyclic operator with cyclic vector x∈ H; and for λ∈B(T), k(λ) will denote the vector of Has defined in (1.1). 2. Description of B(T)and Ba(T). The proofs of Proposition 2.1 and Lemma 2.2 are similar to the ones for the cyclic subnormal operators (see [2], and [8]); we include them for completeness. A complete description of B(T) is given by the following result. 304 A. Bourhim Proposition 2.1. Let λ∈C; the following statements are equivalent. (i)λ∈B(T). (ii)λ∈Γ(T). (iii) ker (T−λ)∗is one dimensional. Proof. First, note that if (T−λ)∗u= 0 for some u∈ H, then for every polynomial p, we have (2.3) hp(T)x, ui=p(λ)hx, ui. Next, we mention that it suffices to prove the implications (ii)⇒(i) and (i)⇒(iii), since the implication (iii)⇒(ii) can be deduced trivially from the fact that Γ(T) = σp(T∗). Let λ∈B(T); it is clear that k(λ)6= 0 since hx, k(λ)i= 1. On the other hand, for every polynomial p, we have 0 = h(T−λ)p(T)x, k(λ)i =hp(T)x, (T−λ)∗k(λ)i. Since xis a cyclic vector for T, we have (T−λ)∗k(λ) = 0. Hence, λ∈σp(T∗). Now, let u∈ H be such that (T−λ)∗u= 0. It follows then from equation (2.3) that u=hx , uik(λ). Therefore, (i)⇒(iii). Let λ∈Γ(T). Then there is a non-zero vector u∈ H such that (T−λ)∗u= 0. Since xis a cyclic vector of T, it follows from equation (2.3) that hx, ui 6= 0. Hence, p(λ) = hp(T)x, u hx, uiifor every polynomial p. Therefore, λ∈B(T) and k(λ) = u hx,ui, which proves that (ii)⇒(i).  Lemma 2.2. An open subset Oof Cis contained in Ba(T)if and only if it is contained in B(T)and the function λ7−→ kk(λ)kis bounded on compact subsets of O. Proof. Assume that O⊂Ba(T) and let Kbe a compact subset of O. For every y∈ H, the function byis analytic on O; in particular, sup λ∈K |hy, k(λ)i| <+∞. So it follows from the Uniform Boundedness Principle that sup λ∈K kk(λ)k<+∞. Conversely, suppose that O⊂B(T) and the function λ7−→ kk(λ)kis bounded on compact subsets of O. Let y∈ H; then there is a sequence of polynomials (pn)nsuch that lim n→+∞kpn(T)x−yk= 0. And so, for every compact subset Kof O, it follows from the Cauchy-Schwartz inequality that, sup λ∈K |pn(λ)−by(λ)| ≤ sup λ∈K kk(λ)kkpn(T)x−yk. Hence, the function byis analytic on O. Therefore, O⊂Ba(T).  The following gives a complete description of Ba(T). Bounded point evaluations for . . . 305 Theorem 2.3. Ba(T) = <(T∗). Proof. Suppose that λ∈ <(T∗). Then there is a non-zero analytic H–valued function φon some open disk Vcentered at λsuch that (T−µ)φ(µ) = 0 for all µ∈ V. Using the fact that a non-zero analytic H–valued function has isolated zeros, one can assume that the function φhas no zero in V.Hence, V ⊂ σp(T∗) = B(T).As before, it follows from (2.3) that k(µ) = φ(µ) hx, φ(µ)ifor every µ∈ V. Therefore, the function k:V → H is continuous; in particular, the function µ7−→ kk(µ)kis bounded on compact subsets of V. By Lemma 2.2, V ⊂ Ba(T). Thus <(T∗)⊂Ba(T). Conversely, set O=Ba(T) and consider the following H–valued function φ defined on Oby φ(λ) := k(λ), λ ∈O. First, we show that φis analytic on O. Indeed, for every y∈ H and for every λ0∈O, we have lim λ→λ0 hφ(λ), yi−hφ(λ0), yi λ−λ0 = lim λ→λ0 hk(λ), yi−hk(λ0), yi λ−λ0 = lim λ→λ0by(λ)−by(λ0) λ−λ0 =by0(λ0). Hence, for every y∈ H, the function λ7−→ hφ(λ), yiis differentiable on O; therefore, φis analytic on O. On the other hand, φhas no zeros on Oand satisfies the following equation (T∗−λ)φ(λ) = 0 for every λ∈O. This gives O=Ba(T)⊂ <(T∗), and the proof is complete.  Corollary 2.4. The following holds: Ba(T) = {λ∈Γ(T) : σT∗−λ(k(λ)) = ∅} ={λ∈Γ(T) : σT∗(k(λ)) = ∅}. Proof. Since, for every λ∈Ba(T), λis a simple eigenvalue of T∗with corresponding eigenvector k(λ), the proof follows by combining Theorem 2.3 and Theorem 1.9 of [1].  Remark 2.5. (i) Note that Proposition 2.1 and Proposition 2.3 show that both B(T) and Ba(T) are independent of the choice of cyclic vector for T(see Proposition 1.4 of [21]). (ii) Using Theorem 2.3 and Theorem 2.6 of [1], one can easily prove (1.2). 306 A. Bourhim 3. Resolution of question 1.1. The weighted shift operators have proven to be an interesting rich collection of operators providing examples and counterexamples to illustrate many properties of operators. The Allen Shields’s excellent survey [18] contains their basic facts and properties concerning their spectral theory (see also [2]). Throughout this section, Swill denote a unilateral weighted shift operator on Hwith a positive bounded weight sequence (ωn)n≥0, that is Sen=ωnen+1, n ≥0, where (en)n≥0is an orthonormal basis of H. Let (βn)n≥0be the following sequence given by: βn=   ω0...ωn−1if n > 0 1 if n= 0 Set r1(S) = lim n→∞ inf k≥0 βn+k βk1 n , r2(S) = lim inf n→∞ βn1 nand r(S) = lim n→∞ sup k≥0 βn+k βk1 n ; and note that, r1(S)≤r2(S)≤r(S)≤ kSk. The following gives a necessary and sufficient condition for the weighted shift Sto answer affirmatively Question 1.1. Theorem 3.1. The following are equivalent. (i)Ba(S) = Γ(S)\σap(S). (ii)r1(S) = r2(S). Proof. Since σp(S∗)⊂ {λ∈C:|λ| ≤ r2(S)}(see Theorem 9 of [18]), we have <(S∗)⊂O:= {λ∈C:|λ|< r2(S)}. Conversely, consider the following non-zero analytic H–valued function φdefined on Oby φ(λ) := +∞ X n=0 λn βn en. It is easy to see that (S∗−λ)φ(λ) = 0 for every λ∈O. Hence, O={λ∈C:|λ|< r2(S)} ⊂ <(S∗). Therefore, <(S∗) = {λ∈C:|λ|< r2(S)}; by Theorem 2.3, (3.4) Ba(S) = {λ∈C:|λ|< r2(S)}. On the other hand, by [18, Theorems 4 and 6], the spectrum and the approximate point spectrum of Sare given, respectively, by σ(S) = {λ∈C:|λ| ≤ r(S)}and σap(S) = {λ∈C:r1(S)≤ |λ| ≤ r(S)}. Bounded point evaluations for . . . 307 Hence, (3.5) Γ(S)\σap(S) = σ(S)\σap(S) = {λ∈C:|λ|< r1(S)}. And so the proof follows from (3.4) and (3.5).  Now, to give a counterexample to Question 1.1, we only need to produce a weight sequence (ωn)n≥0for which the corresponding weighted shift Ssatisfies r1(S)< r2(S). Example 3.2. Let (Ck)k≥0be the sequence of successive disjoint segments covering the set of non-negative integers Nsuch that each segment Ckcontains k2+kelements. Let R > 1 be a real number and let k∈N; for every n∈Ck we set, ωn=   Rif nlies in the first k2terms of Ck 1 otherwise Hence, the unilateral weighted shift Scorresponding to the weight (ωn)n≥0is bounded and satisfies kSk=Rand r1(S) = 1. On the other hand for every n≥3, there is k(n)≥2 such that n∈Ck(n). Hence, R k(n)−1 P s=1 s2 ≤βnand n≤ k(n) X s=1 (s2+s). Therefore, R(2k(n)−1)(k(n)−1) 2(k(n)+1)(k(n)+2) ≤βn1 n. Since lim n→+∞k(n) = +∞, it follows that, R≤lim inf n→+∞βn1 n. We deduce that r1(S) = 1 and r2(S) = kSk=R. Thus, Γ(S)\σap(S) = {λ∈C:|λ|<1}$Ba(S) = {λ∈C:|λ|< R}. The original idea of this construction is due to W. C. Ridge [17]. For other example see [3], where the authors constructed a unilateral weighted shift Sfor which Γ(S)\σap(S) = ∅and Ba(S) = {λ∈C:|λ|<1}. 4. Resolution of question 1.2. If Tpossesses Bishop’s property (β), then we obtain the following. Theorem 4.1. If Tpossesses Bishop’s property (β), then the following are equivalent. (i)Ba(T) = Γ(T)\σap(T). (ii)Ba(T)∩σp(T) = ∅. 308 A. Bourhim Proof. If Ba(T) = Γ(T)\σap(T) then it is clear that Ba(T)∩σp(T) = ∅ since σp(T)⊂σap(T). Conversely, suppose that Ba(T)∩σp(T) = ∅. Since Γ(T)\σap(T)⊂Ba(T)⊂Γ(T), it suffices to prove that Ba(T)∩σap(T) = ∅. Suppose that there is λ∈Ba(T)∩σap(T). It then follows that ran(T−λ) is not closed. Let y∈clran(T−λ)\ran(T−λ); then y6∈ ran(T−λ) and hy, k(λ)i= 0. Therefore, there is a sequence of polynomials (pn)nvanishing at λsuch that lim n→+∞kpn(T)x−yk= 0. Define on U:= Ba(T) the following analytic H–valued functions fand fnby f(µ) = y−by(µ)xand fn(µ) = pn(T)x−pn(µ)x, n ≥0. Since f(λ) = y6∈ ran(T−λ), then f6∈ ran(TU). On the other hand, it is easy to see that fn∈ran(TU) for every n≥0. Now, let Kbe a compact subset of U; we have sup µ∈K kfn(µ)−f(µ)k≤kpn(T)x−yk+ sup µ∈K kpn(µ)−by(µ)xk ≤ kpn(T)x−yk+kxksup µ∈K |pn(µ)−by(µ)| ≤1 + kxksup µ∈K kk(µ)kkpn(T)x−yk. Therefore, fn−→ fin O(U, H). Thus ran(TU) is not closed which contradicts the fact that Tpossess Bishop’s property (β). The proof is complete.  The following is immediate (see Theorem 3.1 of [23]). Corollary 4.2. Suppose that Tpossesses Bishop’s property (β). If σp(T) = ∅, then Ba(T) = Γ(T)\σap(T). The following gives a positive answer to Question 1.2. Theorem 4.3. If Tis hyponormal, then Ba(T) = Γ(T)\σap(T). Proof. Since Tis hyponormal, then for every λ∈C, we have T−λis hyponormal i.e., (4.6) k(T−λ)∗yk ≤ k(T−λ)yk,∀y∈ H. Now, let λ∈Ba(T) and suppose that there is y∈ H such that Ty =λy. For every µ∈Ba(T), we have, λby(µ) = hTy , k(µ)i =hy , T∗k(µ)i =hy , µk(µ)i =µby(µ). Hence, the analytic function byis identically zero on Ba(T). On the other hand, it follows from (4.6) and Proposition 2.1 that y=αk(λ) for some α∈C. Therefore, by(λ) = αkk(λ)k2= 0 gives y= 0. Thus, Ba(T)∩σp(T) = ∅. By Theorem 4.1, the proof is complete.  Bounded point evaluations for . . . 309 In [16], M. Raphael has shown that two quasisimilar cyclic subnormal operators have equal approximate point spectra. The following generalizes M. Raphael’s result to cyclic hyponormal operators. Suppose that H1and H2 are Hilbert spaces. Recall that two operators T1∈ L(H1) and T2∈ L(H2) are said to be quasisimilar if there exist two bounded linear transformations X:H1→ H2and Y:H2→ H1injectives and having dense range such that XT1=T2Xand T1Y=Y T2. Corollary 4.4. Suppose that H1and H2are Hilbert spaces, and let T1∈ L(H1) and T2∈ L(H2). If T1and T2are quasisimilar cyclic hyponormal operators, then σap(T1) = σap(T2). Proof. In view of Theorem 1.5 of [21] and Theorem 4.3, we have Γ(T1)\σap(T1) = Γ(T2)\σap(T2). Hence, σ(T1)\σap(T1) = σ(T2)\σap(T2). Since σ(T1) = σ(T2) (see Theorem 2 of [6], and also Corollary 2.2 of [23]), we deduce that σap(T1) = σap(T2).  Remark 4.5. Theorem 4.3 and Corollary 4.4 can be extended with no extra effort to the class of operators satisfying the following. (i)Tpossesses Bishop’s property (β). (ii) ker(T−λ)⊂ker(T−λ)∗for every λ∈C. Immediate other examples of operators satisfying (i) and (ii) are provided by M–hyponormal and p–hyponormal operators (see [14] and [9]). 5. examples and comments. In this section, we start by proving the following result that we will need throughout. Proposition 5.1. If H=W{k(λ) : λ∈Ba(T)}, then σp(T) = ∅. Moreover, if H=W{k(λ) : λ∈G}for some connected component Gof Ba(T), then the following hold. (i) cl(G)⊂σT(y)for every non-zero element y∈ H. (ii)For every y∈ H,σT(y)is a connected set. (iii)σ(T)is a connected set. Proof. Suppose that H=W{k(λ) : λ∈Ba(T)}, and Ty =λy for some λ∈C and y∈ H. For every µ∈Ba(T)\{λ}, we have, λby(µ) = hTy, k(µ)i =hy, T ∗k(µ)i =µby(µ). Hence, the analytic function byis identically zero on Ba(T); this means that hy, k(λ)i= 0 for every λ∈Ba(T). Since H=W{k(λ) : λ∈Ba(T)}, we have 316 A. Bourhim [8] J. B. Conway, The Theory of Subnormal Operators, volume 36 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, R.I, 1991. [9] B. P. Duggal, p-hyponormal operators satisfy Bishop’s condition (β), Integral Equations Operator Theory 40 (2001), 436–440. [10] J. K. Finch, The single valued extension property on a Banach space, Pacific J. Math. 58 (1975), 61–69. [11] P. R. Halmos, A Hilbert Space Problem Book, Springer-Verlag, New York, 1982. [12] D. E. Herrero, On multicyclic operators, Integral Equations and Operator Theory, 1/1 (1978), 57-102. [13] K. B. Laursen and M. M. Neumann, An Introduction to Local Spectral Theory, London Mathematical Society Monograph New Series 20 (2000). [14] M. Putinar, Hyponormal operators are subscalar, J. Operator Theory, 12 (1984), 385-395. [15] C. R. Putnam, An inequality for the area of hyponormal spectra, Math. Z. 116 (1970), 323-330. [16] M. Raphael, Quasisimilarity and essential spectra for subnormal operators, Indiana Univ. Math. J. 31 (1982), 243-246. [17] W. C. Ridge, Approximate point spectrum of a weighted shift, Trans. Amer. Math. Soc. 147 (1970), 349-356. [18] A. L. Shields, Weighted shift operators and analytic function theory, in Topics in Operator Theory, Mathematical Surveys, N0 13 (ed. C. Pearcy), pp. 49-128. American Mathematical Society, Providence, Rhode Island, 1974. [19] J. G. Stampfli, On Hyponormal and Toeplitz operators, Math. Ann. 183 (1969), 328-336. [20] T. T. Trent, H2(µ)Spaces and bounded point evaluations, Pac. J. Math., 80 (1979), 279-292. [21] L. R. Williams, Bounded point evaluations and local spectra of cyclic hyponormal operators, Dynamic Systems and Applications 3(1994), 103-112. [22] L. R. Williams, Subdecomposable operators and rationally invariant subspaces, Operator theory: Adv. Appl., 115 (2000), 297-309. [23] L. Yang, Hyponormal and subdecomposable operators, J. Functional Anal. 112 (1993), 204217. Received November 2001 Revised September 2002 A. Bourhim The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy E-mail address:[email protected] Current address: A. Bourhim D´epartement de Math´ematiques, Universit´e Mohamed V, B.P. 1014, Rabat, Morocco E-mail address:[email protected]