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Stability of the bases and frames reproducing kernels in model spaces. (English) Zbl 1101.30036
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 S. V. Hruscev, N. K. Nikolski and 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 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 J. Ortega-Cerdá and 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 [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.

MSC:
30D55 \(H^p\)-classes (MSC2000)
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
47A45 Canonical models for contractions and nonselfadjoint linear operators
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