A Note on Norm-Attaining Properties for Frame Operators

Abstract

This paper examines the relationship between frame operators, dual frames, and norm attainability in Hilbert spaces. Frames are basis-like systems that span a vector space while allowing linear dependency, enabling the achievement of desirable properties not available with orthonormal bases. Frames provide redundant and stable vector representations, which are particularly valuable in signal processing applications. Norm attainability refers to the condition where a frame operator acts as a scalar multiple of a vector. This study explores how frame bounds, redundancy, and tightness influence the norm-attaining properties of frame operators and their duals. Theoretical results are developed to enhance the understanding of norm-attaining operators and offer insights into designing frames with optimal properties for practical applications, especially in signal processing.