Stability of the bases and frames reproducing kernels in model spaces.
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 space \(L^2(0,\pi)\). Moreover, a system of reproducing kernels \(k_{\lambda_n}\) in \(K_{\Theta}^2 \) corresponds to a system of complex exponentials \(e^{i \lambda_n t}\) in \(L^2(0,\pi)\). The complete description of Riesz bases of exponentials was obtained by \textit{S. V. Hruscev, N. K. Nikolski} and \textit{B. S. Pavlov} [``Unconditional bases of exponentials and of reproducing kernels'', Complex Analysis and Spectral Theory, Semin. Leningrad 1979/80, Lect. Notes in Math. 864, 214--335 (1981; Zbl 0466.46018)] in terms of the Helson-Szegö condition. These authors also treated the case of general inner functions and obtained a necessary and sufficient condition under the additional restriction \[ \sup_n|\Theta(\lambda_n)| < 1 \,. \tag{2} \] In this case the system \(\{k_{\lambda_n}\}\) is a basis if and only if the sequence \(\Lambda\) satisfies the Carleson interpolation condition and the Toeplitz operator \(T_{\Theta \overline{B}}\), where \(B\) is the Blaschke product with zeros \(\lambda_n\), is invertible. For the case when \(\Theta\) and \(\Lambda\) satisfies (2) a result on stability of the bases under small perturbations of frequencies \(\lambda_n\) was obtained by \textit{E. Fricain} [``Bases of reproducing kernels in model spaces'', J. Oper. Theory 46, No. 3, 517--543 (2001; Zbl 0995.46021)]. Namely, if \(\{k_{\lambda_n}\}\) is a basis in \(K_{\Theta}^2 \), then there is \(\varepsilon > 0\) such that \(\{k_{\mu_n}\}\) is a basis whenever \[ \sup_n \rho (\lambda_n, \mu_n) < \varepsilon , \tag{3} \] where \(\rho (z,w) \) denotes the pseudo-hyperbolic metric in \(\mathbb{C}^+\). A system \(\{h_n\}\) in a Hilbert space \(H\) is a frame if there are positive constant \(A\) and \(B\) such that \[ A\|f\|_H^2 \leq \sum_{n}|(f,h_n)_H|^2 \leq B\|f\|_H^2, \quad f \in H. \] Here, \((f,g)_H\) denotes the inner product in \(H\). If the system \(\{h_n\}\) is a Riesz basis, then it is automatically a frame. If \(\{ k_{\lambda_n}/\|k_{\lambda_n}\|_2\}\) is a frame in \(K_{\Theta}^2\), then \[ A\|f\|_2^2 \leq \sum_{n}|f(\lambda_n)|^2 \|k_{\lambda_n} \|_2^2 \leq B\|f\|_2^2, \quad f \in K_{\Theta}^2. \tag{6} \] A set \(\Lambda\) satisfying (6) is a sampling set for \(K_{\Theta}^2\). Sampling sets for \(K_{\Theta}^2\) with \(\Theta (z) = \exp{(2 \pi i z)}\) were described by \textit{J. Ortega-Cerdá} and \textit{K. Seip} [``Fouries frames'', Ann. Math. 155, 789--806, (2002; Zbl 1015.42023)]. In the paper the author considers an approach to the problem of stability of bases and frames of reproducing kernels bases on estimates of derivatives obtained by the own author in a previous paper [\textit{A. Barnov}, ``Bernstein-type inequalities for shift-coinvariant subspaces and their applications to Carleson embeddings'', J. Funct. Anal. 223, 116--146 (2005; Zbl 1082.46019)]. This makes possible to give unified proofs and generalize results of E. Fricain and W. S.Cohn. Let \(\{ k_{\lambda_n}\}\) be a basis in \(K_{\Theta}^2\) and \(G = \bigcup_nG_n \subset \overline{\mathbb{C}^+}\) satisfy the following properties: (i) There exist positive constants \(c\) and \(C\) such that \[ c \leq \|k_{z_n}\|_2/ \|k_{\lambda_n}\|_2 \leq C, \quad z_n \in G_n; \] (ii) For any \(z_n \in G_n\) the measure \(\nu = \sum_n \delta_{<\lambda_n, z_n>}\) is a Carleson measure where \(\delta_{<\lambda_n, z_n>}\) is the Lebesgue measure of the interval with endpoints \(\lambda_n\) and \(z_n\) and, moreover, the Carleson constants \(M_{\nu}\) of such measures are uniformly bounded with respect to \(z_n\). The main result on stability (applicable to general inner functions \(\Theta\) and \(\Lambda\)) state (Theorem 1.1) that if \(\{ k_{\lambda_n}\}\) is a basis in \(K_{\Theta}^2\), \(p \in (1,2)\), \(1/p + 1/q = 1,\) then for any set \(G\) satisfying (i) and (ii) there exists \(\varepsilon > 0\) such that the systems of reproducing kernels \(\{k_{\mu_n}\}\) is a basis whenever \(\mu_n \in G_n\) and \[ \sup_n \frac{1}{\|k_{\lambda_n}\|_2^2} \int_{< \lambda_n, \mu_n>} \|k_z^2\|_q^{\frac{2p}{p + 1}}|dz| < \varepsilon . \tag{9} \] And, in the case of real ``frequencies'' \(\{t_n\} \subset \mathbb{R}\) and if we put \(d_0(t) = dist(t, \sigma(\Theta))\) where \(\sigma(\Theta)\) is the spectrum of the inner function \(\Theta\), it is shown (Theorem 1.4) that if \(\{k_{t_n}\}\) is a basis (or \(\{k_{t_n}/\|k_{t_n}\|_2\}\) a frame in \(K_{\Theta}^2\)), then there exists \(\varepsilon > 0\) such that \(\{k_{s_n}\}\) is a basis (or \(\{k_{s_n}\|k_{s_n}\|_2\}\) a frame in \(K_{\Theta}^2\)) whenever \[ \int_{<t_n,s_n>}\left( |\Theta'(t)| + \frac{1}{|\Theta ' (t)|d^2_0(t)} \right)dt < \varepsilon \tag{14} \] or \[ |s_n - t_n| < \varepsilon |\Theta'(t_n)|\min(d^2_0(t), |\Theta'(t_n)|^{-2}). \tag{15} \] It is noted that the admissible perturbations in both results depend essentially on the properties of the function \(\Theta \) and the density of its spectrum near the point under consideration.
scientific article; zbMATH DE number 5015293
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