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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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