The Upper Semi-Weylness and Positive Nullity for Operator Matrices

期刊论文
发表日期：

2024年3月25日

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Tengjie Zhang,
Xiaohong Cao,
Jiong Dong

摘要

Let H and K be infinite dimensional separable complex Hilbert spaces and B(K, H) the algebra of all bounded linear operators from K into H. Let $A\in B(H)$ $A \in B (H)$ and $B\in B(K)$ $B \in B (K)$ . 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, we give necessary and sufficient conditions for $M_C$ $M_{C}$ to be an upper semi-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 operator $C\in B(K,H)$ $C \in B (K, H)$ . Meanwhile, we discover the relationship between $n(M_C)$ $n (M_{C})$ and n(A) during the exploration. And we also describe all left invertible operators $C\in B(K,H)$ $C \in B (K, H)$ such that $M_C$ $M_{C}$ is an upper semi-Fredholm operator with $n(M_C)>0$ $n (M_{C}) > 0$ and $\hbox {ind}(M_C)<0$ $ind (M_{C}) < 0$ .