Journal of Operator Theory

Volume 46, Issue 3, Supplementary 2001 pp. 517-543.

Bases of reproducing kernels in model spaces

Authors: Emmanuel Fricain
Author institution: Universite Bordeaux I, UFR Mathematiques/Informatique, 351, Cours de la Liberation, 33405 Talence Cedex, France

Summary: This paper deals with geometric properties of sequences of reproducing kernels related to invariant subspaces of the backward shift operator in the Hardy space

$H^2$ . Let

$\La=(\la_n)_{n\great 1}\subset{\bbb D}$ ,

$\t$ be an inner function in

$H^\infty({\cal L}(E))$ , where

$E$ is a finite dimensional Hilbert space, and

$(e_n)_{n\great 1}$ a sequence of vectors in

$E$ . Then we give a criterion for the vector valued reproducing kernels

$(k_\t( \, \cdot \, ,\la_n)e_n)_{n\great 1}$ to be a Riesz basis for

$K_\t:=H^2(E)\ominus \t H^2(E)$ . Using this criterion, we extend to the vector valued case some basic facts that are well-known for the scalar valued reproducing kernels. Moreover, we study the stability problem, that is, given a Riesz basis

$(k_\t( \, \cdot \, ,\la_n))_{n\great 1}$ , we characterize its perturbations

$(k_\t( \, \cdot \, ,\mu_n))_{n\great 1}$ that preserve the Riesz basis property. For the case of asymptotically orthonormal sequences, we give an effective upper bound for uniform perturbations preserving stability and compare our result with Kade\v {c}'s

$1/4$ -theorem.

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