Duals of a frame in quaternionic Hilbert spaces

TL;DR

This paper extends the theory of frames in quaternionic Hilbert spaces by introducing various duals, and provides explicit formulas for orthogonal projections related to frames and their duals.

Contribution

It introduces different types of duals for frames in quaternionic Hilbert spaces and derives formulas for orthogonal projections using these duals.

Findings

Explicit formulas for orthogonal projections in quaternionic Hilbert spaces.

Introduction of new dual frame types in quaternionic setting.

Application to analysis operator range projection.

Abstract

Frames in a separable quaternionic Hilbert space were introduced and studied in [17] to have more applications. In this paper, we extend the study of frames in quaternionic Hilbert spaces and introduce different types of duals of a frame in separable quaternionic Hilbert spaces. As an application, we give the orthogonal projection of $\ell^2(\HH)$ onto the range of analysis operator of the given frame, in terms of elements of canonical dual frame and elements of the frame in quaternionic Hilbert space. Finally, we give an expression for the orthogonal projection in terms of operators related to the frame and its canonical dual frame in quaternionic Hilbert space.