S-spectrum and Associated Continuous Frames on Quaternionic Hilbert Spaces

TL;DR

This paper investigates the S-spectrum of operators on quaternionic Hilbert spaces and uses it to construct and analyze rank n continuous frames, extending concepts from complex to quaternionic settings.

Contribution

It introduces the S-spectrum for quaternionic operators and applies it to develop and classify continuous frames on quaternionic Hilbert spaces.

Findings

Defined the S-spectrum for quaternionic operators

Constructed rank n continuous frames using the S-spectrum

Established equivalence classes of these frames

Abstract

As needed for the construction of rank $n$ continuous frames on a right quaternionic Hilbert space the so-called S-spectrum of a right quaternionic operator is studied. Using the S-spectrum, as for the case of complex Hilbert spaces, along the lines of the arguments of {\em Ann.Phys.}, {\bf 222} (1993), 1-37., various classes of rank $n$ continuous frames and their equivalencies on a right quaternionic Hilbert space are presented.