幂等算子组合的可逆性

The Invertibility of Combinations of Idempotents

讨论了希尔伯特空间上的两个不同的幂等算子P、Q的组合aP+bQ-cPQ的可逆性问题,利用幂等算子的性质和空间分解的技巧证明了aP+bQ-cPQ的可逆性与系数的选取无关,其中a,b,c∈瓘,ab≠0,a+b-c≠0.而且构造反例说明该结果不能推广到aP+bQ-cPQ-dQP的情形.

Discussed the problem of invertibility of combinations of two different idempotent operators P,Q on a Hilbert space. The invertibility of aP ＋ bQ-cPQ are independent choice of coefficients, where a, b, c ∈ C ,ab ≠ 0 ,a ＋b ≠c which was proved by using the properties of idempotent operators and the techniques of space decomposition. Moreover, some counter examples were constructed to denote that the result can not be generalized to the case aP ＋ bQ-cPQ-dQP .