Cyclic vectors for backward hyponormal weighted shifts

Author:
Shelley Walsh

Journal:
Proc. Amer. Math. Soc. **96** (1986), 107-114

MSC:
Primary 47B37

MathSciNet review:
813821

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Abstract: A unilateral weighted shift on a Hilbert space is an operator such that for some orthonormal basis and weight sequence . If we assume , for all , and let for and , then is unitarily equivalent to on the weighted space of formal power series such that . Regarding as multiplication for on the space , it is shown that, if and is analytic in a neighborhood of the unit disk, then either is cyclic for or is contained in a finite-dimensional -invariant subspace. This was shown--by different methods-- for the unweighted shift operator by Douglas, Shields, and Shapiro [**2**]. It is also shown that every finite-dimensional -invariant subspace is of the form

**[1]**John B. Conway,*Subnormal operators*, Research Notes in Mathematics, vol. 51, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1981. MR**634507****[2]**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**0270196****[3]**Allen L. Shields,*Weighted shift operators and analytic function theory*, Topics in operator theory, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. Math. Surveys, No. 13. MR**0361899****[4]**S. Walsh,*Cyclic vectors for the backward Bergman shift*, Dissertation, Univ. of California, 1984.

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DOI:
https://doi.org/10.1090/S0002-9939-1986-0813821-0

Article copyright:
© Copyright 1986
American Mathematical Society