The structures and decompositions of symmetries involving idempotents

Abstract

Xi ′ an, 710062, P eople ′ s Republic of China. b. College of Xingzhi, Xi ′ an U niversity of F inance and Economics, Xi ′ an, 710038, P eople ′ s Republic of China.Abstract Let H be a separable Hilbert space and P be an idempotent on H. We denote byIn this paper, we first get that symmetries (2P − I)|2P − I| −1 and (P + P * − I)|P + P * − I| −1 are the same. Then we show that Γ P = ∅ if and only if ∆ P = ∅. Also, the specific structures of all symmetries J ∈ Γ P and J ∈ ∆ P are established, respectively. Moreover, we prove that J ∈ ∆ P if and only ifwhere U is a unitary operator from R(P ) ⊥ onto R(P ) with U * P 1 = P * 1 U. Proof. Suppose that J has the following operator matrix form J = J 11 J 12 J * 12 J 22 : R(P ) ⊕ R(P ) ⊥ , where J 11 and J 22 are self-adjoint operators. It follows from the fact JP = (I − P )J that        J 11 = −P 1 J * 12 1 J 11 P 1 = −P 1 J 22 2 J * 12 P 1 = J 22 3 . (2.9) 6 On the other hand, J = J * = J −1 yields J 2 = I, so       