General construction of Reproducing Kernels on a quaternionic Hilbert space

TL;DR

This paper develops a comprehensive theory of reproducing kernels and Hilbert spaces in quaternionic settings, connecting positive operator measures with quaternionic coherent states and extending classical theorems.

Contribution

It introduces a general framework for reproducing kernels on quaternionic Hilbert spaces, including a Naimark extension theorem and illustrative examples from Hermite and Laguerre polynomials.

Findings

Established a general theory of quaternionic reproducing kernels

Proved a Naimark type extension theorem in quaternionic Hilbert spaces

Presented examples from Hermite and Laguerre polynomials

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.