Abstract
In the previous work Bellavita (Complex Anal. Oper. Theory 15: 96, 2021) we found some necessary conditions for the boundedness of the translation operator \(T_\zeta\) in the de Branges space \({{\mathcal {H}}}(E)\). In that case we made use of the Carleson measures for the associated model space. In this work we start from the Pancherel-Polya inequality in the Paley-Wiener space and from the Bernstein inequality in the de Branges space. This different approach allows us to obtain a new condition, in some cases necessary and sufficient, for the boundedness of \(T_\zeta\) in the de Branges space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baranov, A.D.: Differentiation in the Branges spaces and embedding theorems. J. Math. Sci. (N. Y.) 101, 2881–2913 (2000)
Baranov, A.D.: Bernstein’s inequality in the de Branges spaces and embedding theorems. Amer. Math. Soc. Transl. 209, 21–49 (2003)
Baranov, A.D.: Estimate of the \(L^p-\)norms of derivatives in spaces of entire functions. J. Math. Sci. (N. Y.) 129, 3927–3943 (2005)
Baranov, A.D., Bommier-Hato, H.: De Branges spaces and Fock spaces. Complex Var. Elliptic Equ. 63, 907–930 (2018)
Bellavita, C.: Necessary conditions for boundedness of translation operator in de Branges spaces. Complex Anal. Oper. Theory 15, 96 (2021)
De Branges, L.: Hilbert spaces of entire functions. N.J. Prentice Hall, Englewood Cliffs (1968)
Dyakonov, K.M.: Entire functions of exponential type and model subspaces in \(H^p\). J. Math. Sci. (N. Y.) 71, 2222–2233 (1994)
Dyakonov, K.M.: Differentiation in star-invariant subspaces I. Boundedness and compactness. J. Funct. Anal. 192, 364–386 (2002)
Dyakonov, K.M.: Differentiation in star-invariant subspaces II. Schatten class criteria. J. Funct. Anal. 192, 387–409 (2002)
Garnett, J.: Bounded Analytic Functions. Springer, New York (2007)
Kaltenbäck, M., Woracek, H.: Pólya class theory for Hermite-Biehler functions of finite order. J. London Math. Soc. 2(68), 338–354 (2003)
Levin, B.Y.: Distribution of zeros of entire functions. Translations of mathematical monographs. American Mathematical Society, Chicago (1980)
Levin, B.Y., Lyubarskii, Y., Sodin, M., Tkachenko, V.: Lectures on entire functions. American Mathematical Society, Chicago (1996)
Remling, C.: Schrödinger operators and de Branges spaces. J. Funct. Anal. 196, 395–426 (2002)
Romanov, R.: Canonical systems and de Branges spaces. arXiv: Complex Variables, (2014)
Seip, K.: Interpolation and sampling in spaces of analytic functions. University Lecture Series, American Mathematical Society, Providence (2004)
Yosida, K.: Functional analysis. Springer Science, Berlin Heidelberg (1995)
Acknowledgements
I thank Professor M. Peloso for suggesting the problem and for listening to me for many hours. I thank also the anonymous referee for the huge improvements suggested.
Funding
No funding was received to assist with the preparation of this manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Bellavita, C. Necessary and sufficient condition for boundedness of translation operator in de Branges spaces. Collect. Math. 74, 795–815 (2023). https://doi.org/10.1007/s13348-022-00373-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13348-022-00373-6