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.

Keywords: Invariant subspaces, Bergman spaces, Bergman shift, reproducing kernels, Bergman type kernels

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