The minimal and maximal symmetries for $J$-contractive projections

TL;DR

This paper characterizes symmetries that make a projection $J$-contractive, identifies extremal symmetries, and establishes formulas relating various projections, advancing the understanding of $J$-contractive projections in operator theory.

Contribution

It provides a detailed characterization of symmetries for $J$-contractive projections and identifies their minimal and maximal elements, introducing new formulas connecting different projections.

Findings

Characterization of symmetries $J$ for $J$-contractive projections

Identification of minimal and maximal symmetries $J$

Formulas relating different projection operators

Abstract

In this paper, we firstly character the structures of symmetries $J$ such that a projection $P$ is $J$-contractive. Then the minimal and maximal elements of the symmetries $J$ with $P^{\ast}JP\leqslant J$(or $JP\geqslant0)$ are given. Moreover, some formulas between $P_{(2I-P-P^{\ast})^{+}}$ $(P_{(2I-P-P^{\ast})^{-}})$ and $P_{(P+P^{\ast})^-}$ $(P_{(P+P^{\ast})^+})$ are established.