
This paper deals with geometric properties of sequences of reproducing kernels related to de-Branges spaces. If $b$ is a nonconstant function in the unit ball of $H^\infty$ , and $T_b$ is the Toeplitz operator, with symbol $b$, then the de-Branges space, $H(b)$, associated to $b$, is defined by $H(b)=(Id−T_b^* T_b)^{1/2}H^2$ , where $H^2$ is the Hardy space of the unit disk. It is equiped with the inner product such that $(Id −T_b^* T_b)^{1/2}$ is a partial isometry from $H^2$ onto $H(b)$. First, following a work of Ahern-Clark, we study the problem of orthogonal basis of reproducing kernels in $H(b)$. Then we give a criterion for sequences of reproducing kernels which form an unconditionnal basis in their closed linear span. As far as concerns the problem of complete unconditionnal basis in $H(b)$, we show that there is a dichotomy between the case where $b$ is an extreme point of the unit ball of $H^\infty$ and the opposite case.
Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces), de Branges spaces, Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces, reproducing kernels., Riesz bases, [MATH] Mathematics [math], de-Branges spaces, Reproducing kernels, reproducing kernels, Banach algebras of differentiable or analytic functions, \(H^p\)-spaces, Linear operators on function spaces (general), Spaces of bounded analytic functions of one complex variable, Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces), de Branges--Rovnyak spaces, Analysis
Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces), de Branges spaces, Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces, reproducing kernels., Riesz bases, [MATH] Mathematics [math], de-Branges spaces, Reproducing kernels, reproducing kernels, Banach algebras of differentiable or analytic functions, \(H^p\)-spaces, Linear operators on function spaces (general), Spaces of bounded analytic functions of one complex variable, Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces), de Branges--Rovnyak spaces, Analysis
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