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

Abstract

Motivated by the idea of $J$-frame for a Krein space $\text{K}$, introduced by Giribet \textit{et al.} (J. I. Giribet, A. Maestripieri, F. Martínez Pería, 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 $\text{K}$. In this paper we characterize $J$-orthonormal basis for $\text{K}$ 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.