On Essential Spectra of $2\times 2$ Operator Matrices

Abstract:

      Let $M_X=\left(\begin{array} {cc} A & C\\X& B \end{array}\right)$ be a $2\times 2$ operator matrix acting on the Hilbert space ${\cal H}\oplus{\cal K}$. For given $A\in B(H)$, $B\in B(K)$ and $C\in B(K,H)$ the set $\bigcup_{X\in B({\cal H,\cal K})}\sigma_e(M_X)$ is determined, where $\sigma_e(T)$ denotes the essential spectrum.