关键词: 幂等算子, 可逆算子, 算子矩阵

Abstract:

Let P and Q be two idempotents on a Hilbert space H. The  set ω(P) of generalized idempotent operators with respect to P is defined by ω(P)={A ∈B(H): A²=α A+β P, AP=PA=A, P²=P, for some α, β ∈C}. In this note, the author proves that the invertibility of A-B is completely determined by the invertibility of  P-Q,  and R(AB) is closed if and only if  R(PQ) is closed for arbitrary A ∈ω(P) and B ∈ω(Q) such that A²=α A + β P, B²=mB+nQ, where β n ≠ 0, α and m are arbitrary.

Key words: Idempotent, Invertibility, Operator matrix