Bases of reproducing kernels in model spaces.
Emmanuel Fricain · HAL (Le Centre pour la Communication Scientifique Directe) · 2001
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 $\Lambda=(\lambda_n)_n$ be a sequence in the unit disc $\mathbb D$, $\Theta$ be an inner function in $H^\infty(L(E))$, where $E$ is a finite dimensional Hilbert space, and $(e_n)_n$ a sequence of vectors in $E$. Then we give a criterion for the vector valued reproducing kernels $(k_Θ (., λ_n)e_n)_n$ to be a Riesz basis for $K_Θ:=H^2(E)\ominus \Theta H^2(E)$. 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 $(k_Θ (., λ_n))_n$ , we characterize its perturbations $(k_Θ (., µ n))_n$ 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 Kadeč's 1/4-theorem.