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.

Keywords: model subspaces of $H^2$, Riesz bases, reproducing kernels, stability

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