Metric, Schauder and Operator-Valued Frames

TL;DR

This paper introduces and studies various generalized frame concepts for metric and Banach spaces, establishing their properties, stability, and connections with Lipschitz-free Banach spaces, operator theory, and group representations.

Contribution

It develops new notions of metric, Schauder, and operator-valued frames, providing characterizations, stability results, and links to Lipschitz-free Banach spaces and group representations.

Findings

Every separable metric space admits a metric frame.

Characterizations of metric frames are established.

A generalized operator-valued frame unifying known Hilbert space frames is introduced.

Abstract

Notion of frames and Bessel sequences for metric spaces have been introduced. This notion is related with the notion of Lipschitz free Banach spaces. \ It is proved that every separable metric space admits a metric $\mathcal{M}_d$-frame. Through Lipschitz-free Banach spaces it is showed that there is a correspondence between frames for metric spaces and frames for subsets of Banach spaces. Several characterizations of metric frames are obtained. Stability results are also presented. Non linear multipliers are introduced and studied. This notion is connected with the notion of Lipschitz compact operators. Continuity properties of multipliers are discussed. For a subclass of approximated Schauder frames for Banach spaces, characterization result is derived using standard Schauder basis for standard sequence spaces. Duals of a subclass of approximate Schauder frames are completely described. Similarity of this class is characterized and interpolation result is derived using orthogonality. A dilation result is obtained. A new identity is derived for Banach spaces which admit a homogeneous semi-inner product. Some stability results are obtained for this class. A generalization of operator-valued frames for Hilbert spaces are introduced which unifies all the known generalizations of frames for Hilbert spaces. This notion has been studied in depth by imposing factorization property of the frame operator. Its duality, similarity and orthogonality are addressed. Connections between this notion and unitary representations of groups and group-like unitary systems are derived. Paley-Wiener theorem for this class are derived.