J-Frame Sequences in Krein Space

TL;DR

This paper investigates conditions under which projections of $J$-frames in Krein spaces remain $J$-frames and introduces the concept of $J$-frame sequences, extending frame theory to Krein spaces.

Contribution

It establishes sufficient conditions for $J$-frames to be preserved under projections and introduces $J$-frame sequences in Krein spaces, expanding the theoretical framework.

Findings

Proved conditions for $J$-frames to remain $J$-frames after projection.

Introduced and studied properties of $J$-frame sequences.

Extended frame theory concepts from Hilbert to Krein spaces.

Abstract

Let $\{f_n:n\in\mathbb{N}\}$ be a $J$-frame for a Krein space ${\textbf{\textit{K}}}$ and $P_M$ be a $J$-orthogonal projection from ${\textbf{\textit{K}}}$ onto a subspace $M$. In this article we find sufficient conditions under which $\{P_M(f_n):n\in\mathbb{N}\}$ is a $J$-frame for $P_M\textbf{\textit{K}}$ and $\{(I-P_M)f_n\}_{n\in{\mathbb{N}}}$ is a $J$-frame for $(I-P_M)\textbf{\textit{K}}$. We also introduce $J$-frame sequence for a Krein space ${\textbf{\textit{K}}}$ and study some properties of $J$-frame sequence analogues to Hilbert space frame theory.