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Bases of reproducing kernels in model spaces. (English) Zbl 0995.46021

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 \(\Lambda=(\lambda_{n})_{n\geq 1}\subset {\mathbb{D}}\), \(\Theta\) be an inner function in \(H^{\infty}({\mathcal L}(E))\), where \(E\) is a finite dimensional Hilbert space, and \((e_{n})_{n\geq 1}\) a sequence of vectors in \(E\). Then we give a criterion for the vector reproducing kernels \((k_{\Theta}(\cdot, \lambda_{n})e_{n})_{n\geq 1}\) to be a Riesz basis for \(K_{\Theta}:=H^{2}(E)\ominus\Theta H^{2}(E)\). Using this criterion, we extend to the vector valued case some basic facts that are well-known for the scalar reproducing kernels. Moreover, we study the stability problem, that is, given a Riesz basis \((k_{\Theta}(\cdot,\lambda_{n}))_{n\geq 1}\), we characterize its perturbations \((k_{\Theta}(\cdot,\mu_{n}))_{n\geq 1}\) that preserves the Riesz basis property. For the case of asymptotically orthonormal sequences, we give an effective upper bound for the uniform perturbations preserving stability and compare our result with Kadeč \(1/4\)-theorem.

MSC:

46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
30D55 \(H^p\)-classes (MSC2000)
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
42C15 General harmonic expansions, frames
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