The Residual Spectrum and the Continuous Spectrum of Upper Triangular Operator Matrices

Guojun Hai, Alatancang Chen

Let $\Cal H$ and $\Cal K$ be separable infinite dimensional Hilbert spaces. We denote by $M_C$ the $2\times2$ upper triangular operator matrix acting on $\Cal H\oplus\Cal K$ of the form $M_C=\big(\smallmatrix A&C\\0&B\endsmallmatrix)\endbig$. For given operators $A\in\Cal B(\Cal H)$ and $B\in\Cal B(\Cal K)$, the sets $\bigcup\limits_{C\in\Cal B(\Cal K, \Cal H)}{\sigma_r(M_C)}$ and $\bigcup\limits_{C\in\Cal B(\Cal K, \Cal H)}{\sigma_c(M_C)}$ are characterized, where $\sigma_r(\cdot)$ and $\sigma_c(\cdot)$ denote the residual spectrum and the continuous spectrum, respectively.