Tight J-frames in Krein space and the associated J-frame potential

TL;DR

This paper introduces and characterizes $ta-J$-tight frames in Krein spaces, generalizes the $J$-frame potential concept, and identifies conditions for minimal potential in $ta-J$-tight frames.

Contribution

It defines $ta-J$-tight frames in Krein spaces, characterizes $J$-orthonormal bases via $ta-J$-Parseval frames, and extends the $J$-frame potential theory.

Findings

Krein spaces have abundant $ta-J$-Parseval frames.

Necessary and sufficient conditions for sum of $ta-J$-Parseval frames to be a $ta-J$-Parseval frame.

Conditions for minimal $J$-frame potential in $ta-J$-tight frames.

Abstract

Motivated by the idea of $J$-frame for a Krein space $\textbf{\textit{K}}$, introduced by Giribet \textit{et al.} (J. I. Giribet, A. Maestripieri, F. Mart\'inez Per\'{i}a, P. G. Massey, \textit{On frames for Krein spaces}, J. Math. Anal. Appl. (1), {\bf 393} (2012), 122--137.), we introduce the notion of $\zeta-J$-tight frame for a Krein space $\textbf{\textit{K}}$. In this paper we characterize $J$-orthonormal basis for $\textbf{\textit{K}}$ in terms of $\zeta-J$-Parseval frame. We show that a Krein space is richly supplied with $\zeta-J$-Parseval frames. We also provide a necessary and sufficient condition when the linear sum of two $\zeta-J$-Parseval frames is again a $\zeta-J$-Parseval frame. We then generalize the notion of $J$-frame potential in Krein space from Hilbert space frame theory. Finally we provided a necessary and sufficient condition for a $J$-frame potential of the corresponding $\zeta-J$-tight frame to be minimum.