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Continuity and realization of multiplicative maps between RKHS and their cyclicity preserving properties

Part of: Special classes of linear operators Topological algebras, normed rings and algebras, Banach algebras Spaces and algebras of analytic functions Linear function spaces and their duals

Published online by Cambridge University Press:  08 September 2025

Mohana Rahul Nandan Open the ORCID record for Mohana Rahul Nandan [Opens in a new window]  and
Sukumar Daniel Open the ORCID record for Sukumar Daniel [Opens in a new window]
Mohana Rahul Nandan*
Affiliation:
Department of Mathematics, Indian Institute of Technology Hyderabad , Kandi, Sangareddy 502284, India e-mail: suku@math.iith.ac.in
Sukumar Daniel
Affiliation:
Department of Mathematics, Indian Institute of Technology Hyderabad , Kandi, Sangareddy 502284, India e-mail: suku@math.iith.ac.in
*
e-mail: ma20resch11003@iith.ac.in
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Abstract

Motivated by the study of multiplicative linear functionals in reproducing kernel Hilbert space (RKHS) with normalized complete Pick kernel, we define and study the multiplicative linear map between two RKHS. We identify the conditions under which such maps are continuous. Additionally, we prove that any unital cyclicity-preserving linear map is multiplicative. Conversely, we also characterize when a multiplicative linear map is unital cyclicity preserving. These results serve as a generalization of the Gleason–Kahane–Żelazko theorem to the setting of multiplicative maps between two RKHS. We present the composition operator as a natural class of examples of multiplicative linear maps on an RKHS. We also prove that every continuous multiplicative linear operator can be realized as a composition operator on various analytic Hilbert spaces over the unit disc $\mathbb {D}.$

Keywords

Reproducing Kernel Hilbert spaceGleason–Kahane–Zelazko Theoremmultiplicative functionalalgebra homomorphismautomatic continuity

MSC classification

Primary: 46E22: Hilbert spaces with reproducing kernels (= 30H15: Nevanlinna class and Smirnov class
Secondary: 46H40: Automatic continuity 30H10: Hardy spaces 30H20: Bergman spaces, Fock spaces 47B48: Operators on Banach algebras

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Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 14
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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