Extension of frames and bases -- II

TL;DR

This paper extends the theory of operator-valued frames to continuous settings over measure spaces, exploring duality, similarity, orthogonality, stability, and special characterizations, with concrete formulas for finite-dimensional cases.

Contribution

It introduces a continuous version of operator-valued frames and analyzes their properties, including duality and stability, with specific characterizations for locally compact groups.

Findings

Characterization of continuous operator-valued frames on measure spaces

Derivation of variation, dimension, and trace formulas in finite dimensions

Analysis of duality, similarity, and orthogonality properties

Abstract

Operator-valued frame ($G$-frame), as a generalization of frame is introduced by Kaftal, Larson, and Zhang in \textit{Trans. Amer. Math. Soc.}, 361(12):6349-6385, 2009 and by Sun in \textit{J. Math. Anal. Appl.}, 322(1):437-452, 2006. It has been further extended in the paper arXiv:1810.01629 [math.OA] 3 October 2018, so as to have a rich theory on operator-valued frames for Hilbert spaces as well as for Banach spaces. The continuous version has been studied in this paper when the indexing set is a measure space. We study duality, similarity, orthogonality and stability of this extension. Several characterizations are given including a notable characterization when the measure space is a locally compact group. Variation formula, dimension formula and trace formula are derived when the Hilbert space is finite dimensional.