Stability of the bases and frames reproducing kernels in model spaces.
type: Item
instance of: scholarly article
title: Stability of the bases and frames reproducing kernels in model spaces.
zbMATH Open document ID: 1101.30036
publication date: 2006-03-24T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116258
review text: Let \(\Theta\) be an inner function in the upper half-plane \(\mathbb{C^+}\), i.e., \(\lim_{y \to 0^+}|\Theta( x + iy)| = 1\) for almost every \(x \in \mathbb{R}\) with respect to the Lebesgue measure. To each inner function \(\Theta\) is associated the subspace \(K_{\Theta}^2 = H^2 \ominus \Theta H^2\) of the Hardy space \(H^2\) in \(\mathbb{C^+}\). These subspaces play a prominent role both in function and operator theory. For instance, as proved by P.D. Lax, any subspace of \(H^2\) coinvariant with respect to the semigroup of shifts \((U_t)_{t \geq 0} \), \(U_tf(x)= e^{itx}f(x)\), is of the form \(K_{\Theta}^2 \) for some inner function \(\Theta\). Subspaces \(K_{\Theta}^2 \) are often called model subspaces due to the relation with the Sz.-Nagy-Foias model for contractions in a Hilbert spaces. For \(\lambda \in \mathbb{C^+}\), the function \[ k_{\lambda} (z) = \frac{i}{2\pi}\cdot \frac{1 - \overline{\Theta(\lambda)}\Theta(z)} {z - \overline{\lambda}} \] is the reproducing kernel of the space \(K_{\Theta}^2\) corresponding to the point \(\lambda\). A system of vectors \(\{h_n\}\) in Hilbert space \(H\) is said to be a Riesz basis if \(\{h_n\}\) is an image of an orthogonal basis under a bounded and invertible linear operator on \(H\). In the paper, the author is concerned with the sets of complex numbers \(\Lambda = \{\lambda\}_n\) such that the normalized kernels \(k_{\lambda_n}/ \|k_{\lambda_n}\|_2\) constitute a Riesz basis in \(K_{\Theta}^2 \) for a given \(\Theta\), in short, \(k_{\lambda_n}\) is a basis for \(K_{\Theta}^2 \). In particular, he is interested in stability of the property: given a basis \(k_{\lambda_n}\) the problem is determine what small perturbations \(\mu_n \) of \(\lambda_n\) are admissible so that the property to be basis be preserved. One important example motivates the interest of this problem. Consider the inner function \(\Theta(z) = \exp(2\pi i z)\). Then the Fourier image of the model subspace \(K_{\Theta}^2 \) coincides with the s…
Mathematics Subject Classification ID: 30D55
Mathematics Subject Classification ID: 30F45
Mathematics Subject Classification ID: 46E22
Mathematics Subject Classification ID: 47A45
zbMATH DE Number: 5015293
zbMATH Keywords: inner function
zbMATH Keywords: shift-coinvariant subspace
zbMATH Keywords: reproducing kernel
zbMATH Keywords: Riesz basis
zbMATH Keywords: frame
zbMATH Keywords: stability
author: A. D. Baranov
reviewed by: Fernando Pérez-González
MaRDI profile type: Publication
OpenAlex ID: W279428099
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DOI: 10.5802/AIF.2165
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5015293