Duality for Frames in Krein Spaces

TL;DR

This paper explores the duality theory of $J$-frames in Krein spaces, establishing their properties, and characterizing special cases like tight and Parseval $J$-frames, thus extending frame theory to indefinite inner-product spaces.

Contribution

It introduces and characterizes duality for $J$-frames in Krein spaces, including the concepts of tight and Parseval $J$-frames, advancing the understanding of frames in indefinite inner-product spaces.

Findings

Duality for $J$-frames is established.

Tight and Parseval $J$-frames are characterized.

Maximal uniformly definite subspaces are linked to $J$-frames.

Abstract

A $J$-frame for a Krein space $\mathcal{H}$ is in particular a frame for $\mathcal{H}$ (in the Hilbert space sense). But it is also compatible with the indefinite inner-product of $\mathcal{H}$, meaning that it determines a pair of maximal uniformly definite subspaces, an analogue to the maximal dual pair associated to an orthonormal basis in a Krein space. This work is devoted to study duality for $J$-frames in Krein spaces. Also, tight and Parseval $J$-frames are defined and characterized.