Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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)$.


References [Enhancements On Off] (What's this?)

  • [1] W. B. Arveson, A density theorem for operator algebras, Duke Math. J. 34 (1967), 635-647. MR 0221293 (36:4345)
  • [2] James A. Deddens, Ralph Gellar and Domingo A. Herrero, Commutants and cyclic vectors, Proc. Amer. Math. Soc. 43 (1974), 169-170. MR 0328643 (48:6985)
  • [3] R. G. Douglas, H. S. Shapiro and A. L. Shields, On cyclic vectors of the backward shift, Bull. Amer. Math. Soc. 73 (1967), 156-159. MR 0203465 (34:3316)
  • [4] -, Cyclic vectors and invariant subspaces for the backward shift operator, Ann. Inst. Fourier (Grenoble) 20 (1970), fasc. 1, 37-76. MR 0270196 (42:5088)
  • [5] Mary R. Embry, Strictly cyclic operator algebras on a Banach space, Pacific J. Math. 45 (1973), 443-452. MR 0318922 (47:7468)
  • [6] R. Gellar, Cyclic vectors and parts of the spectrum of a weighted shift, Trans. Amer. Math. Soc. 146 (1969), 69-85. MR 0259642 (41:4277b)
  • [7] D. A. Herrero, Eigenvectors and cyclic vectors for bilateral weighted shifts, Rev. Un. Mat. Argentina 26 (1972), 26-41. MR 0336395 (49:1170)
  • [8] R. V. Kadison, On the orthogonalization of operator representations, Amer. J. Math. 78 (1955), 600-621. MR 0072442 (17:285c)
  • [9] Alan Lambert, Strictly cyclic operator algebras, Pacific J. Math. 39 (1971), 717-726. MR 0310664 (46:9762)
  • [10] -, Subnormality and weighted shifts, J. London Math. Soc. (2) 14 (1976), 476-480. MR 0435915 (55:8866)
  • [11] N. K. Nikolskiĭ, The invariant subspaces of certain completely continuous operators, Vestnik Leningrad Univ. (7) 20 (1965), 68-77. (Russian) MR 0185444 (32:2911)
  • [12] -, Invariant subspaces of weighted shift operators, Math. USSR-Sb. 3 (1967), 159-176.
  • [13] E. A. Nordgren, H. Radjavi and P. Rosenthal, On density of transitive algebras, Acta Sci. Math. (Szeged) 30 (1969), 175-179. MR 0253061 (40:6276)
  • [14] M. Rabindranathan, On cyclic vectors of weighted shifts, Proc. Amer. Math. Soc. 44 (1974), 293-299. MR 0350491 (50:2983)
  • [15] H. Radjavi and P. Rosenthal, Invariant subspaces, Springer-Verlag, Berlin-Heidelberg-New York, 1973. MR 0367682 (51:3924)
  • [16] B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, Akadémiai Kiadó, Budapest, 1970. MR 0275190 (43:947)
  • [17] B. S. Yadav and S. Chatterjee, On a partial solution of the transitive algebra problem, Acta Sci. Math. (Szeged) (to appear). MR 576958 (81m:47015)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47B37

Retrieve articles in all journals with MSC: 47B37


Additional Information

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

American Mathematical Society