Linear Algebra Seminar

Let $$M_{C}=\begin{bmatrix}A & C\\                                     0 & B\\                                  \end{bmatrix}: \mathbb{D}(M_{C})\subset X\times X\rightarrow X\times X $$ be a \(2\times 2\) unbounded upper triangular operator matrix in the complex Hilbert space \(X\times X\).  We investigate the  conditions under which \(\sigma(M_{C})=\sigma(A)\cup\sigma(B)\)  holds in the diagonally dominant (\(\mathbb{D}(M_{C})=\mathbb{D}(A)\times\mathbb{D}(B)\)) and upper dominant case (\(\mathbb{D}(M_{C})=\mathbb{D}(A)\times\mathbb{D}(C)\)). Some  necessary and sufficient conditions are obtained.