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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^∗JP≤ J(or JP≥0) are given. Moreover, some formulas between P_(2I-P-P^∗)^+ (P_(2I-P-P^∗)^-) and P_(P+P^∗)^- (P_(P+P^∗)^+) are established.arXiv:1810.05381v2 fatcat:v3klllcrzbaovizi233opdttzm