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.