广义幂等算子差的可逆性

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): A²=αA+βP, AP=PA=A,P²=P,∨α, β∈C}. 对任意A ∈ω(P), B∈ω(Q)使得A²=αA +βP, B²=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.

Invertibility of Differences of Two Generalized |Idempotent Operators

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.