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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Cyclic vectors of Lambert's weighted shifts


Authors: B. S. Yadav and S. Chatterjee
Journal: Proc. Amer. Math. Soc. 80 (1980), 100-104
MSC: Primary 47B37
DOI: https://doi.org/10.1090/S0002-9939-1980-0574516-9
MathSciNet review: 574516
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Abstract: Let $ B(H)$ denote the Banach algebra of all bounded linear operators on an infinite-dimensional separable complex Hilbert space H, and let $ {l^2}$ be the Hilbert space of all square-summable complex sequences $ x = \{ {x_0},{x_1},{x_2}, \ldots \} $. For an injective operator A in $ B(H)$ and a nonzero vector f in H, put $ {w_m} = \left\Vert {{A^m}f} \right\Vert / \left\Vert {{A^{m - 1}}f} \right\Vert,m = 1,2, \ldots .$ The operator $ {T_{A,f}}$ on $ {l^2}$, defined by $ {T_{A,f}}(x) = \{ {w_1}{x_1},{w_2}{x_2}, \ldots \} $, is called a weighted (backward) shift with the weight sequence $ \{ {w_m}\} _{m = 1}^\infty $. This paper is concerned with the investigation of the existence of cyclic vectors of $ {T_{A,f}}$. Also it is shown that if A satisfies certain nice conditions, then every transitive subalgebra of $ B(H)$ containing $ {T_{A,f}}$ coincides with $ B(H)$.


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DOI: https://doi.org/10.1090/S0002-9939-1980-0574516-9
Keywords: Banach algebras, cyclic vectors, cyclic sets, Hilbert space, invariant subspaces, strictly cyclic operator algebras, transitive algebras, weighted shifts
Article copyright: © Copyright 1980 American Mathematical Society