Spectra of 2 × 2 Upper Triangular Operator Matrices

Abstract

The spectra of the $2\times 2$ upper triangular operator matrix $M_C =\big({A\ C\atop 0\ B}\big)$ acting on a Hilbert space $H_1\oplus H_2$ are investigated. We obtain a necessary and sufficient condition of $\sg(M_C)=\sg(A)\cup\sg(B)$ for every $C\in\calB(H_2,H_1)$, in terms of the spectral properties of two diagonal elements $A$ and $B$ of $M_C$. Also, the analogues for the point spectrum, residual spectrum and continuous spectrum are further presented. Moveover, we construct some examples illustrating our main results. In particular, it is shown that the inclusion $\sg_r(M_C)\subseteq\sg_r(A)\cup \sg_r(B)$ for every $C\in\calB(H_2,H_1)$ is not correct in general. Note that $\sg(T)$ (resp.\,$\sg_r(T)$) denotes the spectrum (resp. residual spectrum) of an operator $T$, and $\calB(H_2,H_1)$ is the set of all bounded linear operators from $H_2$ to $H_1$.