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General construction of Reproducing Kernels on a quaternionic Hilbert space

Research paper by K. Thirulogasanthar, S. Twareque Ali

Indexed on: 17 Jan '16Published on: 17 Jan '16Published in: Mathematical Physics



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.