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• W2070291238 abstract "Let B(H) denote the Banach algebra of all bounded linear operators on an infinite-dimensional separable complex Hilbert space H, and let 12 be the Hilbert space of all square-sumnmable complex sequences x = (xO, xI, x2, . . . }. For an injective operator A in B(H) and a nonzero vector f in H, put wm = IIA mfII/IIA mIf , m = 1, 2, .... The operator TA,S on 12, defined by TA,j(x) = {wIxI, w2x2,... ), is called a weighted (backward) shift with the weight sequence {W,, }m 1. This paper is concerned with the investigation of the existence of cyclic vectors of TAf. Also it is shown that if A satisfies certain nice conditions, then every transitive subalgebra of B(H) containing TAJ coincides with B(H). 1. Let H be an infinite-dimensional separable complex Hilbert space with an orthonormal basis {em1'=o and let B(H) be the Banach algebra of all (bounded linear) operators from H into H. If {wmr =1 is a bounded sequence of nonzero complex numbers, the unique operator T on H defined by Teo = 0 and Tem = wmem -l, m = 1, 2, . . ., is called the (unilateral backward) weighed shift with the weight sequence { wm , }' =. A concrete realisation of the weighted shift is obtained by considering the Hilbert space 12 of all square-summable complex sequences x = {x0, XI, x2, ... }. The weighted shift T on 12 appears as T{x0, xl, x2, . . . } = {wIx1, w2x2, . . . A vector x0 E H is called a cyclic vector of T if V =0{ T'xo} = H. The existence of cyclic vectors of weighted shifts has been the subject of investigation by many authors; see, for example, Douglas, Shapiro and Shields [3], [4], Gellar [6], Herrero [7], Deddens, Gellar and Herrero [2], Nikolskil [11], [12] and Rabindranathan [14]. We say that an operator A is power-bounded if IA l < 8 for all n = 1, 2, 3, . . ., where 8 is a constant. 2. Let A E B(H) and suppose that A is injective. For a vectorf =# 0 in H, let TAJf be the weighted shift on 12 with the weight sequence {wmrn I =, where Wm = IlA mfllf/All lfA (ll) The subnormality of the weighted shifts TAf has been recently studied by Lambert [10]. In this paper we exhibit the existence of cyclic vectors of such weighted shifts under nice conditions on the operator A. Received by the editors May 1, 1979 and, in revised form, June 13, 1979. AMS (MOS) subject classifications (1970). Primary 46E20, 46L20, 47A15, 47B99, 30A78; Secondary 47A60, 47C99." @default.
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• W2070291238 date "1980-01-01" @default.
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• W2070291238 title "Cyclic vectors of Lambert’s weighted shifts" @default.
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• W2070291238 doi "https://doi.org/10.1090/s0002-9939-1980-0574516-9" @default.
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