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The Feichtinger conjecture for reproducing kernels in model subspaces. (English) Zbl 1230.30039

A sequence of unit vectors \(\{h_n\}\) in a separable Hilbert space \(\mathcal{H}\) is said to be a Bessel sequence if for some constant \(C>0\) and every \(h\in \mathcal{H}\), \[ \sum_{n}|\langle h, h_n\rangle|^2\leq C\|h\|^2. \] Further, \(\{h_n\}\) is called a Riesz basic sequence if there exists a constant \(A>0\) such that \[ A^{-1}\sum_{n}|c_n|^2\leq\Big\| \sum_{n}c_nh_n\Big\|^2\leq A\sum_{n}|c_n|^2 \] for every choice of complex numbers \(\{c_n\}\). The Feichtinger conjecture states that every Bessel sequence is the union of finitely many Riesz basic sequences.
The validity of this conjecture has been verified for several classical Hilbert spaces (e.g. Hardy and Bergman spaces). The authors consider spaces of the form \(K_{\Theta}=H^2\ominus\Theta H^2\), where \(H^2\) is the usual Hardy space on the unit disk and \(\Theta\) is an inner function. They prove that under specific assumptions on \(\Theta\), every sequence of normalized reproducing kernels for \(K_\Theta\) is a Bessel sequence that splits into finitely many Riesz basic sequences.

MSC:

30J05 Inner functions of one complex variable
30H10 Hardy spaces
30E05 Moment problems and interpolation problems in the complex plane
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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References:

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