Cyclic vectors of Lambert’s weighted shifts

Yadav, B. S.; Chatterjee, S.

doi:10.1090/S0002-9939-1980-0574516-9

Cyclic vectors of Lambert’s weighted shifts
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by B. S. Yadav and S. Chatterjee PDF

Proc. Amer. Math. Soc. 80 (1980), 100-104 Request permission

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 ${w_m} = \left \| {{A^m}f} \right \| / \left \| {{A^{m - 1}}f} \right \|,m = 1,2, \ldots .$ 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

William B. Arveson, A density theorem for operator algebras, Duke Math. J. 34 (1967), 635–647. MR 221293
James A. Deddens, Ralph Gellar, and Domingo A. Herrero, Commutants and cyclic vectors, Proc. Amer. Math. Soc. 43 (1974), 169–170. MR 328643, DOI 10.1090/S0002-9939-1974-0328643-9
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 203465, DOI 10.1090/S0002-9904-1967-11695-1
R. G. Douglas, H. S. Shapiro, and A. L. Shields, Cyclic vectors and invariant subspaces for the backward shift operator, Ann. Inst. Fourier (Grenoble) 20 (1970), no. fasc. 1, 37–76 (English, with French summary). MR 270196, DOI 10.5802/aif.338
Mary R. Embry, Strictly cyclic operator algebras on a Banach space, Pacific J. Math. 45 (1973), 443–452. MR 318922, DOI 10.2140/pjm.1973.45.443
Ralph Gellar, Operators commuting with a weighted shift, Proc. Amer. Math. Soc. 23 (1969), 538–545. MR 259641, DOI 10.1090/S0002-9939-1969-0259641-X
Domingo A. Herrero, Eigenvectors and cyclic vectors for bilateral weighted shifts, Rev. Un. Mat. Argentina 26 (1972/73), 24–41. MR 336395
Richard V. Kadison, On the orthogonalization of operator representations, Amer. J. Math. 77 (1955), 600–620. MR 72442, DOI 10.2307/2372645
Alan Lambert, Strictly cyclic operator algebras, Pacific J. Math. 39 (1971), 717–726. MR 310664, DOI 10.2140/pjm.1971.39.717
Alan Lambert, Subnormality and weighted shifts, J. London Math. Soc. (2) 14 (1976), no. 3, 476–480. MR 435915, DOI 10.1112/jlms/s2-14.3.476
N. K. Nikol′skiĭ, The invariant subspaces of certain completely continuous operators, Vestnik Leningrad. Univ. 20 (1965), no. 7, 68–77 (Russian, with English summary). MR 0185444

Invariant subspaces of weighted shift operators

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Eric A. Nordgren, Heydar Radjavi, and Peter Rosenthal, On density of transitive algebras, Acta Sci. Math. (Szeged) 30 (1969), 175–179. MR 253061
M. Rabindranathan, On cyclic vectors of weighted shifts, Proc. Amer. Math. Soc. 44 (1974), 293–299. MR 350491, DOI 10.1090/S0002-9939-1974-0350491-4
Heydar Radjavi and Peter Rosenthal, Invariant subspaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77, Springer-Verlag, New York-Heidelberg, 1973. MR 0367682, DOI 10.1007/978-3-642-65574-6
Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190
B. S. Yadav and S. Chatterjee, On a partial solution of the transitive algebra problem, Acta Sci. Math. (Szeged) 42 (1980), no. 1-2, 211–215. MR 576958