Journal of Operator Theory

Volume 62, Issue 1, Summer 2009 pp. 199-214.

Cyclic vectors and invariant subspaces for Bergman and Dirichlet shifts

Authors: Eva A. Gallardo-Gutierrez

$1$ , Jonathan R. Partington

$2$ , and Dolores Segura

$3$
Author institution:

$1$ Departamento de Matematicas, Universidad de Zaragoza e IUMA, Plaza San Francisco s/n, 50009 Zaragoza, Spain

$2$ School of Mathematics, Univ. of Leeds, Leeds LS2 9JT, U.K.

$3$ C. Regidor, Bloque 6, 2C, 11630 Arcos de la Frontera

$Cadiz$ , Spain

Summary: It is shown that the invariant subspaces for the Bergman and Dirichlet shifts on the right half-plane correspond to the common invariant subspaces of the right shift operators on certain weighted Lebesgue spaces on the half-line. As a particular instance, the corresponding result for invariant subspaces of multipliers induced by weak-star generators of

$\mathcal {H}^{\infty}(\mathbb{D})$ on weighted Bergman spaces of the unit disc is deduced. Finally, cyclic vectors for the Bergman and Dirichlet shifts are also studied.

Keywords: Invariant subspaces, shift operator, Bergman spaces, Dirichlet spaces, cyclic vectors.

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