幂等算子组合的指数及Fredholm性

The index and Fredholmness of combinations of idempotent operators

如果P,Q是希尔伯特空间上的两个不同的幂等算子,2006年杜鸿科等证明了线性组合aP+bQ的可逆性与系数的选取无关,其中a,b∈C,ab≠0,a+b≠0.该文将上述结果推广为aP+bQ-cPQ的指数与Fredholm性与系数的选取无关,其中a,b,c∈C,ab≠0,a+b-c≠0.而且构造反例说明不能推广到aP+bQ-cPQ-dQP的情形.

If P、Q are two different idempotent operators on Hilbert space,Du,etc.in 2006 proved that the invertiblilty of linear combinations of aP＋bQ is independent of the choice of coefficients,where a,b∈C,ab≠0,a＋b≠0.In the present note,this results are strengthened by showing that the index and Fredholmness of aP＋bQ-cPQ are also independent of the choice of coefficients,where a,b,c∈C,ab≠0,a＋b-c≠0.Moreover,some examples are constructed to denote that the above result can not be generalized to the case aP＋bQ-cPQ-dQP.