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General construction of reproducing kernels on a quaternionic Hilbert space

Department of Computer Science and Software Engineering, Concordia University, 1455 De Maisonneuve Blvd. West, Montréal, Québec, H3G 1M8, Canada

E-mail Address: santhar@gmail.com

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and

S. Twareque Ali

Department of Mathematics and Statistics, Concordia University, Montréal, Québec, H3G 1M8, Canada

E-mail Address: twareque.ali@concordia.ca

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https://doi.org/10.1142/S0129055X17500179Cited by:2 (Source: Crossref)

Abstract

A general theory of reproducing kernels and reproducing kernel Hilbert spaces on a right quaternionic Hilbert space is presented. Positive operator-valued measures and their connection to a class of generalized quaternionic coherent states are examined. A Naimark type extension theorem associated with the positive operator-valued measures is proved in a right quaternionic Hilbert space. As illustrative examples, real, complex and quaternionic reproducing kernels and reproducing kernel Hilbert spaces arising from Hermite and Laguerre polynomials are presented. In particular, in the Laguerre case, the Naimark type extension theorem on the associated quaternionic Hilbert space is indicated.

Keywords:

AMSC: 46E22, 47B32, 47B34, 81R30

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