Duality principles in Hilbert‐Schmidt frame theory

Abstract

This paper addresses duality relations in HS‐frame theory. By introducing the notion of HS‐R‐dual sequence, some duality relations and related results are obtained. We prove that a sequence is an HS‐frame (HS‐frame sequence, HS‐Riesz basis) if and only if its HS‐R‐dual sequence is an HS‐Riesz sequence (HS‐frame sequence, HS‐Riesz basis) and characterize the (unitary) equivalence between two HS‐frames in terms of their HS‐R‐duals and transition matrices, respectively. We characterize HS‐R‐duals and prove that, given an HS‐frame, among all its dual HS‐frames, only the canonical dual admits minimal‐norm HS‐R‐dual. And using HS‐R‐duals, we also characterize dual HS‐frame pairs.