Discussion on Matrices Fixed Nullity in Complement Problem of Operator Matrices

期刊论文
发表日期：

2024年5月15日

查看全文

Tengjie Zhang,
Xiaohong Cao,
Jiong Dong

摘要

Let H and K be separable infinite-dimensional Hilbert spaces, and let $A\in B(H)$ $A \in B (H)$ and $B\in B(K)$ $B \in B (K)$ be given operators. We denote by $M_C$ $M_{C}$ the operator acting on $H\oplus K$ $H \oplus K$ of the form $M_C=\left( \begin{array}{cc}A&{}C\\ 0&{}B\\ \end{array}\right) $ $M_{C} = (\begin{matrix} A & C \\ 0 & B \end{matrix})$ . In this paper, some necessary and sufficient conditions are obtained for $M_C$ $M_{C}$ to be a Fredholm operator with $n(M_C)>0$ $n (M_{C}) > 0$ and $\hbox {ind}(M_C)<0$ $ind (M_{C}) < 0$ for some left invertible or invertible operator $C\in B(K,H)$ $C \in B (K, H)$ . Meanwhile, for the nullity of $M_C$ $M_{C}$ , we discuss the relationship between $n(M_C)$ $n (M_{C})$ and n(A) by different method. As the application of above results, the weak properties of Weyl’s theorem for upper triangular operator matrices are explored.