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

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 $P^{\ast }JP⩽J$(or $JP⩾0)$ 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.