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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 $ T$ on a Hilbert space $ H$ is an operator such that $ T{e_n} = {w_n}{e_{n + 1}}$ for some orthonormal basis $ \left\{ {{e_n}} \right\}_{n = 0}^\infty $ and weight sequence $ \left\{ {{w_n}} \right\}_{n = 0}^\infty $. If we assume $ {w_n} > 0$, for all $ n$, and let $ \beta \left( n \right) = {w_0} \cdots {w_{n - 1}}$ for $ n > 0$ and $ \beta \left( 0 \right) = 1$, then $ T$ is unitarily equivalent to $ f \mapsto zf$ on the weighted space $ {H^2}\left( \beta \right)$ of formal power series $ \sum\nolimits_{n = 0}^\infty {\hat f\left( n \right){z^n}} $ such that $ \sum\nolimits_{n = 0}^\infty {{{\left\vert {\hat f\left( n \right)} \right\vert}^2}{{\left[ {\beta \left( n \right)} \right]}^2} < \infty } $. Regarding $ T$ as multiplication for $ z$ on the space $ {H^2}\left( \beta \right)$, it is shown that, if $ {w_n} \uparrow 1$ and $ f$ is analytic in a neighborhood of the unit disk, then either $ f$ is cyclic for $ {T^ * }$ or $ f$ is contained in a finite-dimensional $ {T^ * }$-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 $ {T^ * }$-invariant subspace is of the form

$\displaystyle {\left( {{{\left( {z - {\alpha _1}} \right)}^{{n_1}}} \cdots {{\l... ... {z - {\alpha _k}} \right)}^{{n_k}}}{H^2}\left( \beta \right)} \right)^ \bot },$

for some $ {\alpha _1}, \ldots ,{\alpha _k}$ in the unit disk and $ {n_1}, \ldots ,{n_k}$ positive integer.

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

  • [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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Article copyright: © Copyright 1986 American Mathematical Society