Slice Regular Composition Operators on Quaternionic Fock Spaces via Matrix Realization

Abstract

We characterize the boundedness and compactness of slice regular composition operators between quaternionic Fock spaces for the full range \(0<p,q<\infty\), without assuming that the composition symbol preserves a fixed complex slice. As applications of the same method, we also obtain corresponding criteria for weighted composition operators and for products of Volterra-type integral operators with slice regular composition operators. The main tool is a fixed-slice matrix realization of the regular product, which represents slice regular composition on a fixed complex slice through a holomorphic \(2\times 2\) matrix functional calculus. This representation reveals a genuinely quaternionic rigidity phenomenon: boundedness imposes affine restrictions on the eigenvalue functions of the associated matrix symbol rather than on the original symbol itself. In particular, the original symbol need not be affine, and affine eigenvalue functions alone do not characterize boundedness.

arXiv comment39 pages; Keywords: Quaternionic Fock space; slice regular composition; matrix functional calculus; Fock--Carleson measures; eigenvalue rigidity; weighted composition operators