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Cauchy–de Branges Spaces, Geometry of Their Reproducing Kernels and Multiplication Operators

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Abstract

Cauchy–de Branges spaces are Hilbert spaces of entire functions defined in terms of Cauchy transforms of discrete measures on the plane and generalizing the classical de Branges theory. We consider extensions of two important properties of de Branges spaces to this, more general, setting. First, we discuss geometric properties (completeness, Riesz bases) of systems of reproducing kernels corresponding to the zeros of certain entire functions associated to the space. In the case of de Branges spaces they correspond to orthogonal bases of reproducing kernels. The second theme of the paper is a characterization of the density of the domain of multiplication by z in Cauchy–de Branges spaces.

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Acknowledgements

The author is grateful to Artur Nicolau for useful discussions concerning Frostman shifts, to Vladimir Shemyakov for the help with numerical experiments and to Antonio Rivera for the discussions of the material of Sect. 7.

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  1. Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia

    Anton Baranov

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  1. Anton Baranov
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Correspondence to Anton Baranov.

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The results of Sects. 3, 4, 5 and 6 were obtained with the support of the Russian Science Foundation grant 19-71-30002. The results of Sects. 7 and 8 were obtained with the support of Ministry of Science and Higher Education of the Russian Federation, agreement No 075-15-2021-602.

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Baranov, A. Cauchy–de Branges Spaces, Geometry of Their Reproducing Kernels and Multiplication Operators. Milan J. Math. 91, 97–130 (2023). https://doi.org/10.1007/s00032-023-00378-1

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