On $J$-frames related to maximal definite subspaces

TL;DR

This paper introduces an extended definition of $J$-frames in Krein spaces, allowing for maximal definite subspaces that are not necessarily uniformly definite, thereby broadening the scope of $J$-frame concepts.

Contribution

It proposes a new, more general definition of $J$-frames in Krein spaces that includes non-uniformly definite subspaces, expanding the types of sequences considered as $J$-frames.

Findings

Extended $J$-frames include non-uniformly definite subspaces.

Complete $J$-orthogonal sequences can be interpreted as $J$-frames.

$J$-orthogonal Schauder bases are also considered as $J$-frames.

Abstract

A definition of frames in Krein spaces is proposed which extends the concept of $J$-frames defined by J.I. Giribet et al., J. Math. Anal. Appl. ${\textbf{393}}$ (2012), 122-137. The principal difference consists in the fact that a $J$-frame is related to maximal definite subspaces $\mathcal{M}_\pm$ which are not assumed to be uniformly definite. The latter allows one to extend the collection of $J$-frames. In particular, some complete $J$-orthogonal sequences and $J$-orthogonal Schauder bases can be interpreted as $J$-frames.