广义幂等算子差的可逆性

Invertibility of Differences of Two Generalized Idempotent Operators

设P,Q为Hilbert空间H上的幂等算子,关于算子P的广义幂等算子类ω(P)定义为ω(P)={A∈B(H):A^2=αA+βP,AP=PA=A,P^2=P,(?)α,β∈C}.对任意A∈ω(P),B∈ω(Q)使得A^2=αA+βP,B^2=mB+nQ,βn≠0,得到了如下的结论:值域R(PQ)是闭的充要条件是值域R(AB)是闭的;如果P-Q是可逆的,则A-B是可逆的.

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.