Journal of Operator Theory

Volume 51, Issue 1, Winter 2004 pp. 181-200.

Reproducing kernels and invariant subspaces of the Bergman shift

Authors: George Chailos
Author institution: University of Tennessee, Knoxville TN 37920, USA and Intercollege, Makedonitissas Ave, 1700 Nicosia, Cyprus

Summary: In this article we consider index

$1$ invariant subspaces

$M$ of the operator of multiplication by

$\zeta(z)=z$ ,

$M_\zeta$ , on the Bergman space

$\Ber$ of the unit disc.\ It turns out that there is a positive sesquianalytic kernel

$l_\lambda$ defined on

${\bbb D} \times {\bbb D}$ which determines

$M$ uniquely. We set

$\sigma(M_\zeta^*\Mprep$ to be the spectrum of

$M_\zeta^*$ restricted to

$M^\perp$ , and we consider a conjecture due to Hedenmalm which states that if

$M \neq \Ber$ , then

$\rank l_\lambda$ equals the cardinality of

$\sigma(M_\zeta^*\Mprep$ . In this direction we show that

$cardinality\, \left(\sigma(M_\zeta^*\Mprep \cap {\bbb D}\right) \leq \rank l_\lambda \leq cardinality\, \sigma(M_\zeta^*\Mprep$ and furthermore, we resolve the conjecture in the case of zero based invariant subspaces. Moreover, we describe the structure of

$l_\lambda$ for finite zero based invariant subspaces.

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