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 $L^2_a(\mathbb{D})$ of the unit disc.\ It turns out that there is a positive sesquianalytic kernel $l_\lambda$ defined on ${\mathbb D} \times {\mathbb D}$ which determines $M$ uniquely. We set $\sigma(M_\zeta^*|M^\perp)$ to be the spectrum of $M_\zeta^*$ restricted to $M^\perp$, and we consider a conjecture due to Hedenmalm which states that if $M \neq L^2_a(\mathbb{D})$, then $\rm{rank}\ l_\lambda$ equals the cardinality of $\sigma(M_\zeta^*|M^\perp)$. In this direction we show that $cardinality\, \left(\sigma(M_\zeta^*|M^\perp \cap {\mathbb D}\right) \leq \rm{rank}\ l_\lambda \leq cardinality\, \sigma(M_\zeta^*|M^\perp)$ 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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