A local spectral condition for strong compactness with some applications to bilateral weighted shifts

arXiv:1208.3245v1 [math.FA] 15 Aug 2012 A LOCAL SPECTRAL CONDITION FOR STRONG COMPACTNESS WITH SOME APPLICATIONS TO BILATERAL WEIGHTED SHIFTS MIGUEL LACRUZ AND MAR´ IA DEL PILAR ROMERO DE LA ROSA Abstract. An algebra of bounded linear operators on a Banach space is said to be strongly compact if its unit ball is precompact in the strong operator topology, and a bounded linear operator on a Banach space is said to be strongly compact if the algebra with identity generated by the operator is strongly compact. Our interest in this notion stems from the work of Lomonosov on the existence of invariant subspaces. We provide a local spectral condition that is sufficient for a bounded linear operator on a Banach space to be strongly compact. This condition is then applied to describe a large class of strongly compact, injective bilateral weighted shifts on Hilbert spaces, extending earlier work of Fern´andez-Valles and the first author. Further applications are also derived, for instance, a strongly compact, invertible bilateral weighted shift is constructed in such a way that its inverse fails to be a strongly compact operator. 1. Introduction Let B(X) denote the algebra of bounded linear operators on a complex Banach space X. A subalgebra R ⊆ B(X) is said to be strongly compact if its unit ball {R∈ R :kRk ≤ 1} is precompact in the strong operator topology, and an operator T∈ B(X) is said to be strongly compact if the algebra with identity generated by Tis strongly compact. Our interest in this notion stems from the work of Lomonosov [7] on the existence of invariant subspaces for essentially normal operators on Hilbert space. He showed that if Tis an essentially normal operator on a Hilbert space such that both its commutant {T}′ and the commutant {T∗}′of its adjoint fail to be strongly compact algebras then Thas a nontrivial invariant subspace, and moreover, if both Tand T∗fail to be strongly compact operators then Thas a nontrivial hyperinvariant subspace. Lomonosov, Radjavi and Troitsky [8] showed more recently that if Tis an operator such that its commutant {T}′is a localizing algebra and the commutant {T∗}′of its adjoint is a strongly compact algebra, then the adjoint T∗has an invariant subspace. A characterization of strongly compact, normal operators was given by Lomonosov, Rodr´ıguez-Piazza, and the first author [5] by a suitable application of the spectral theorem. Necessary and sufficient conditions were also provided for a unilateral weighted shift to be strongly compact in terms of the sliding products of its weights. Some applications were 2000 Mathematics Subject Classification. Primary 47B07; Secondary 47B37, 47L10. Keywords and phrases: Strongly compact operator, local spectral condition, bilateral weighted shift. This work was partially supported by Ministerio de Ciencia e Innovaci´on under project MTM 2009-08934, and by Junta de Andaluc´ıa under grant FQM-3737 . 1 2 MIGUEL LACRUZ AND MAR´ IA DEL PILAR ROMERO DE LA ROSA derived, for instance, the restriction of a strongly compact operator to an invariant subspace need not be a strongly compact operator. Prunaru [11] used a particular case of a more general result obtained by Lomonosov [7] to show that for a pure hyponormal, essentially normal operator Ton a Hilbert space, the commutant {T∗}′of its adjoint is a strongly compact algebra. Rodr´ıguez-Piazza and the first author [6] showed that the position operator on the space of square integrable functions with respect to a finite measure of compact support is strongly compact if and only if the restriction of the measure to the exterior boundary of its support is purely atomic. (A similar result was obtained earlier on by Froelich and Marsalli [4] within the framework of function algebras.) Further applications were derived, for instance, the weakly closed algebra generated by a strongly compact normal operator need not be a strongly compact algebra. A classification of operator algebras was provided by Marsalli [9]. He gave some sufficient conditions for an algebra of operators to be strongly compact. He showed that if an operator has a set of eigenvectors that generate a dense linear manifold, then the operator is strongly compact, and moreover, if all the corresponding eigenvalues have finite multiplicity, then the commutant of the operator is strongly compact. Quite recently, Fern´andez-Valles and the first author [3] applied this condition to test strong compactness for several classes of operators, namely, Ces`aro operators, bilateral weighted shifts, and composition operators. A condition of a different nature seems to be needed to prove strong compactness in absence of eigenvalues. The purpose of this paper is to provide a local spectral condition that is sufficient for an operator on a Banach space to be strongly compact. The condition requires from the operator that the origin must lie in the interior of its full spectrum and that there must be a spanning set of vectors where the local spectral radius is smaller than the distance from the origin to the boundary of the full spectrum. This condition can be applied to operators with no eigenvalues and it fits like a glove to bilateral weighted shifts. As an application, we describe a large class of strongly compact, bilateral weighted shifts on Hilbert spaces, completing the work of Fern´andez-Valles and the first author [3]. Then, we restrict our attention to invertible bilateral weighted shifts and we derive a condition for such operators to have a strongly compact inverse. Next, we show that if an invertible bilateral weighted shift and its inverse are strongly compact operators then the algebra generated by both of them is strongly compact. Finally, we give an example of a strongly compact, invertible bilateral weighted shift whose inverse fails to be strongly compact. 2. A local spectral condition for strong compactness Let Tdenote an operator on a complex Banach space Xand let σ(T) denote its spectrum. The spectral radius of Tis defined as r(T) = max{|λ|:λ∈σ(T)}. The spectral radius formula provides the alternative expression r(T) = lim n→∞ kTnk1/n. A LOCAL SPECTRAL CONDITION FOR STRONG COMPACTNESS 3 The local spectral radius of Tat a vector x∈Xis defined as r(x, T ) = lim sup n→∞ kTnxk1/n. It is clear that r(x, T )≤r(T) for every x∈X. Daneˇs [2] showed that the set of vectors x∈Xthat satisfy the strict inequality r(x, T )< r(T) is of the first category. See also the paper of M¨uller [10] for more information on this remarkable fact and other related results. Let Gdenote the unbounded connected component of C\σ(T). The full spectrum of Tis the set η(σ(T)) = C\G. It is plain that σ(T)⊆η(σ(T)) and that C\η(σ(T)) is connected. Also, it follows from the maximum modulus principle that if pis any polynomial then max{ |p(λ)|:λ∈σ(T)}= max{ |p(λ)|:λ∈η(σ(T))}. Recall that S⊆Xis a spanning set if Sgenerates a dense linear subspace of X. It turns out that a subalgebra R ⊆ B(X) is strongly compact if and only if there is a spanning set S⊆Xsuch that {Rx :R∈ R,kRk ≤ 1}is precompact for every x∈S. The main result on this paper relies on this simple fact. See the paper of Lomonosov, Rodr´ıguez-Piazza and the first author [5] for a proof of it, and see also the paper of Marsalli [9] for the same result under a slightly different formulation. Theorem 1. Let Tbe an operator on a complex Banach space Xwith 0∈int η(σ(T)) and suppose that there is a spanning set S⊆Xsuch that r(x, T )< d(0, ∂η(σ(T))) for every x∈S. Then the operator Tis strongly compact. Proof. First, consider the positive distance d=d(0, ∂η(σ(T))) = min{|λ|:λ∈∂η(σ(T))}. Since 0 ∈int η(σ(T)), we have {λ∈C:|λ| ≤ d} ⊆ η(σ(T)). It follows from the above remarks and the spectral mapping theorem that for any polynomial pwe have max{|p(λ)|:|λ|=d} ≤ max{ |p(λ)|:λ∈η(σ(T))} = max{ |p(λ)|:λ∈σ(T)} =r(p(T)) ≤ kp(T)k. This inequality allows us to control the size of the coefficients for a complex polynomial p(λ) = a0+a1λ+···+amλm. Indeed, using Cauchy’s integral formula for the derivatives of order n= 0,1,...,m gives an=p(n)(0) n!=1 2πi Z|λ|=d p(λ) λn+1 dλ, and from here we get the estimate |an| ≤ kp(T)k dn. Take a vector x∈S. We need to show that the set {p(T)x:pis a polynomial,kp(T)k ≤ 1} is precompact. Choose some c > 0 with r(x, T )< c < d. Since r(x, T ) = lim sup kTnxk1/n 4 MIGUEL LACRUZ AND MAR´ IA DEL PILAR ROMERO DE LA ROSA as n→ ∞, there is an n0such that kTnxk ≤ cnwhenever n≥n0. Let ε > 0 and choose an n1≥n0such that ∞ X n=n1+1 c dn< ε. Finally, consider the finite dimensional subspace F= span {x, T x, . . . , T n1x}. We claim that p(T)x∈F+εBX. This is obvious if m≤n1. Otherwise, p(T)xcan be expressed as the sum of two terms, p(T)x= n1 X n=0 anzn+ m X n=n1+1 anzn. The first term belongs to F, while the second term satisfies the estimate      m X n=n1+1 anTnx     ≤ m X n=n1+1 |an| · kTnxk ≤ m X n=n1+1 c dn< ε, which completes the proof of our claim.  It is easy to see that strong compactness is preserved under similarities, that is, if Tis a strongly compact operator, Cis invertible, and T=C−1RC then Ris strongly compact. Notice that the assumption in Theorem 1 is also preserved under similarities. More precisely, if Tsatisfies the condition in Theorem 1, Cis invertible, and T=C−1RC, then σ(T) = σ(R), {Cx :x∈S}is a spanning set, and r(Cx, R)≤r(x, T )< d(0, ∂η(σ(T))) = d(0, ∂η(σ(R))) for every x∈S, so that Ralso satisfies the condition in Theorem 1. We now recall a couple of definitions from general operator theory. If Tis any operator on a Banach space Xthen the lower bound of Tis defined as m(T) = inf{kT xk:x∈X, kxk ≤ 1}. We also consider the quantity r1(T) = sup n≥1 m(Tn)1/n = lim n→∞ m(Tn)1/n. It turns out that if Tis invertible then kT−1k=m(T)−1and r(T−1) = r1(T)−1. 3. Applications to bilateral weighted shifts Let Wbe an injective bilateral weighted shift on an infinite dimensional, separable complex Hilbert space H, that is, W en=wnen+1, where (en) is an orthonormal basis of H, the weight sequence (wn) is bounded, nruns through the integers, and wn6= 0 for every n. We refer to the survey by Allen Shields [12] for information on the spectral parts of weighted shifts. See also the paper of Bourhim [1] for local spectral properties of weighted shifts. Now consider the quantities r−(W) = lim n→∞ sup k>0 |w−n−k···w−k+1|1/n, r+(W) = lim n→∞ sup k≥0 |wk···wn+k−1|1/n, A LOCAL SPECTRAL CONDITION FOR STRONG COMPACTNESS 5 r− 1(W) = lim n→∞ inf k>0|w−n−k···w−k+1|1/n, r+ 1(W) = lim n→∞ inf k≥0|wk···wn+k−1|1/n, r− 2(W) = lim inf n→∞ |w−1···w−n|1/n, r+ 2(W) = lim inf n→∞ |w0···wn−1|1/n, r− 3(W) = lim sup n→∞ |w−1···w−n|1/n, r+ 3(W) = lim sup n→∞ |w0···wn−1|1/n. Then the following relationships are fulfilled r− 1(W)≤r− 2(W)≤r− 3(W)≤r−(W), r+ 1(W)≤r+ 2(W)≤r+ 3(W)≤r+(W), r(W) = max{r−(W), r+(W)}, r1(W) = min{r− 1(W), r+ 1(W)}. It was shown by Fern´andez-Valles and the first author [3] that if a bilateral weighted shift Wsatisfies the inequality r+ 3(W)< r− 2(W) then its commutant {W}′is strongly compact, and it remained an open question whether or not Wis strongly compact when r− 2(W)≤r+ 3(W). The following result provides an affirmative answer to this question for quite a large class of bilateral weighted shifts. Theorem 2. If a bilateral weighted shift Won an infinite dimensional, separable complex Hilbert space satisfies the inequality r+ 3(W)< r(W)then Wis a strongly compact operator. Proof. The full spectrum of Wis the closed disk of radius r(W) centered at the origin and the orthornormal basis (ek) is a spanning subset of H. Thus, it suffices to show that r(ek, W ) = r+ 3(W) for each k. This is trivial when k= 0 because kWne0k=|w0···wn−1| for every n≥1,so that r(e0, W ) = lim sup n→∞ kWne0k1/n = lim sup n→∞ |w0···wn−1|1/n =r+ 3(W). Then, suppose that k6= 0 and notice that kWnekk=|wk···wn+k−1|for every n≥1.Now, there are two possibilities. On the one hand, if k > 0 then kWnekk=|w0···wn+k−1| |w0···wk−1|=kWn+ke0k kWke0k, so that r(ek, W ) = lim sup n→∞ kWnekk1/n = lim sup n→∞ kWn+ke0k kWke0k1/n = lim sup n→∞ kWn+ke0k1/n = lim sup n→∞ kWn+ke0k1/(n+k)(n+k)/n =r(e0, W ). On the other hand, if k < 0 and n > −kthen kWnekk=|wk···w−1| · |w0···wn+k−1|=kW−kekk · kWn+ke0k, so that r(ek, W ) = lim sup n→∞ kWnekk1/n = lim sup n→∞ kW−kekk · kWn+ke0k1/n = lim sup n→∞ kWn+ke0k1/n = lim sup n→∞ kWn+ke0k1/n+k(n+k)/n =r(e0, W ), 6 MIGUEL LACRUZ AND MAR´ IA DEL PILAR ROMERO DE LA ROSA as we wanted.  Corollary 3. Let Wbe an invertible bilateral weighted shift on an infinite dimensional, separable complex Hilbert space. If Wsatisfies the inequality r1(W)< r− 2(W)then its inverse W−1is a strongly compact operator. Proof. If Wis an invertible bilateral weighted shift with sequence of weights (wn) then W−1en=1 wn−1 en−1. Consider the unitary operator Udefined by Uen=e−n, so that U∗=U=U−1. Now define V=U∗W−1Uand notice that Vis another bilateral weighted shift, namely, V en=vnen+1, where the sequence of weights (vn) is given by vn=1 w−(n+1) . According with the remarks at the end of the proof of Theorem 1, strong compactness is preserved under similarities. Hence, it suffices to show that Vis strongly compact. Then, a quick computation yields r+ 3(V) = r− 2(W)−1< r1(W)−1=r(V). Theorem 4. Let Wbe an invertible bilateral weighted shift on an infinite dimensional, separable complex Hilbert space. If both Wand W−1are strongly compact then the algebra generated by Wand W−1is strongly compact. Proof. It suffices to show that the set Ck={p(W)ek+q(W−1)ek:p, q are polynomials, q(0) = 0,kp(W) + q(W−1)k ≤ 1} is precompact for every integer k. Since Wand W−1are both strongly compact, the sets C+ k={p(W)ek:pis a polynomial,kp(W)k ≤ 1} C− k={q(W−1)ek:qis a polynomial, q(0) = 0,kq(W−1)k ≤ 1} are both precompact. Notice that for every pair of polynomials p, q with q(0) = 0 we have kp(W)ekk ≤ kp(W)ek+q(W−1)ekkand kq(W−1)ekk ≤ kp(W)ek+q(W−1)ekk. Now it follows that Ck⊆C+ k+C− kso that Ckis precompact, as we wanted.  Corollary 5. Let Wbe an invertible bilateral weighted shift on an infinite dimensional, separable complex Hilbert space. If both inequalities r+ 3(W)< r(W)and r1(W)< r− 2(W) are fulfilled then the algebra generated by Wand W−1is strongly compact. Proof. Since r+ 3(W)< r(W), it follows from Theorem 2 that Wis strongly compact, and since r1(W)< r− 2(W), it follows from Corollary 3 that W−1is also strongly compact. Hence, the desired result is a consequence of Theorem 4.  Example. Consider a bilateral weighted shift Wwhose sequence of weights (wn) satisfies the conditions 1 ≤ |wn| ≤ 2 for each n≥0 and wn= 1 for each n < 0. It is plain that such a weighted shift is bounded and invertible, and that kW−1k= 1. We claim that W−1 fails to be strongly compact, for otherwise the sequence e−n=W−ne0would have a norm A LOCAL SPECTRAL CONDITION FOR STRONG COMPACTNESS 7 convergent subsequence as n→ ∞, a contradiction. Then, the sequence of weights can be chosen in such a way that lim sup n→∞ |w0···wn−1|1/n = 1 and lim n→∞ sup k≥0 |wk···wn+k−1|1/n = 2. There are many possible choices for such a sequence of weights, for instance, if we set wk= 2 whenever 2n≤k≤2n+nand wk= 1 otherwise. Therefore, r+ 3(W) = 1 and r(W) = r+(W) = 2. It follows from Theorem 2 that Wis a strongly compact operator. Summarizing, we constructed a strongly compact bilateral weighted shift whose inverse fails to be strongly compact. Acknowledgement. We are indebted to Fernando Le´on-Saavedra for his kind invitation to visit Universidad de C´adiz at Jerez de la Frontera during the preparation of this manuscript. We had many conversations that made possible the present work. We would like to express our sincere gratitude for his generosity and his hospitality. References [1] A. Bourhim, ‘On the local spectral properties of weighted shift operators’, Studia Math. 163 (2004), no. 1, 41–69. [2] J. Daneˇs, ‘On local spectral radius’, ˇ Casopis Pˇest. Mat. 112 (1987) 177–187. [3] A. Fern´andez-Valles and M. Lacruz, ‘Spectral conditions for strong compactness’, preprint. [4] J. Froelich and M. Marsalli, ‘Operator semigroups, flows, and their invariant sets’, J. Funct. Anal. 115 (1993), no. 2, 454–479. [5] M. Lacruz, V.I. Lomonosov and L. Rodr´ıguez-Piazza, ‘Strongly compact algebras’, Rev. R. Acad. Cien. Serie A. Mat. 100 (2006) 191–207. [6] M. Lacruz, L. Rodr´ıguez-Piazza, ‘Strongly compact normal operators’, Proc. Amer. Math. Soc. 137 (2009), no. 8, 2623–2630. [7] V.I. Lomonosov, ‘A construction of an intertwining operator’, Funktsional. Anal. i Prilozhen. 14 (1980) 67–68. [8] V.I. Lomonosov, H. Radjavi and V.G. Troitsky, ‘Sesquitransitive and localizing operator algebras’, Integr. equ. oper. theory 60 (2008) 405–418. [9] M. Marsalli, ‘A classification of operator algebras’, J. Operator Theory 24 (1990) 155–163. [10] V. M¨uller, ‘Orbits, weak orbits and local capacity of operators’, Integr. equ. oper. theory 41 (2001) 230–253. [11] B. Prunaru, ‘On the commutant of hyponormal operators’, Proc. Amer. Math. Soc. 124 (1996) 3411– 3412. [12] A.L. Shields,‘Weighted shift operators and analytic function theory’, Topics in operator theory, Math. Surveys Amer. Math. Soc. 13 (1974) 49–128. Miguel Lacruz Mar´ıa del Pilar Romero de la Rosa Departamento de An´alisis Departamento de Matem´aticas Universidad de Sevilla Universidad de C´adiz Avenida Reina Mercedes Avenida de la Universidad 41012 Seville (SPAIN) 11402 Jerez de la Frontera (SPAIN) [email protected] [email protected]