关于广义二次算子

设B(H)表示定义在希尔伯特空间H,上的所有有界线性算子的全体.如果A∈B(H)满足二次算子方程A2=αA βP,其中α,β∈C,P是一个非零的幂等算子且AP=PA=A,则称A为广义二次算子.记L(P)为关于幂等算子P的广义二次算子之集.我们用算子谱论的方法研究了L(P)的谱和群逆等相关性质,并推广了R. W. Farebrother和G. Trenkler的结论.

On Generalized Quadratic Operators

Let (B)(H) denote the set of all bounded linear operators on a Hilbert space (H). (A)∈(B)(H) is said to be generalized quadratic operator if A satisfies AP = PA = A and the quadratic equation A2 = αA + βP,where α,β∈ C and P ∈ (B)(H) is nonzero idempotent. Let (L)(P) denote the set of generalized quadratic operators with respect to an idempotent P. In this note,using the technique of operator theory,the spectrum and the group inverse of (L)(P) have been studied. This extends the conclusions of R. W. Farebrother and G. Trenkler.