Oblique dual and g-dual frames in separable quaternionic Hilbert spaces

Quaternionic Hilbert spaces have important applications in quantum physics and the frame theory in quaternionic Hilbert spaces can handle many practical problems in physics. In this paper, we investigate oblique dual and g-dual frames in separable quaternionic Hilbert spaces. We provide a sufficient condition for commutativity of the oblique dual frames in quaternion Hilbert spaces. We present a parametric and algebraic formula for all oblique duals of any given frame in a closed subspace U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {U}$$\end{document} by utilizing a new (right) pre-frame operator. We prove that the canonical oblique dual has a minimum norm representation of elements in U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {U}$$\end{document}. We provide a method for constructing an oblique dual frame from two Bessel sequences. We show that g-duality relation is symmetric and get a necessary condition for g-duality of the sum of two g-dual frames. We also present a parametric expression of all g-dual frames of any given frame and show that approximate dual frame pairs form g-dual frames. We study perturbation-stability of g-dual frames.