INTERSECTIONS OF THE LEFT AND RIGHT ESSENTIAL SPECTRA OF $2\times 2$ UPPER TRIANGULAR OPERATOR MATRICES

Abstract

In this paper, perturbations of the left and right essential spectra of $2\times 2$ upper triangular operator matrix $M_C$ are studied, where $M_C= \left(\begin{smallmatrix} A & C 0 & B \end{smallmatrix}\right)$ is an operator acting on the Hilbert space ${\cal H}\oplus{\cal K }$. For given operators $A$ and $B$, the sets $\bigcap_{C\in B({\cal K},{\cal H})}\sigma_{\rle}(M_C)$ and $\bigcap_{C\in B({\cal K},{\cal H})}\sigma_{\rre}(M_C)$ are determined, where $ \sigma_{\rle}(T)$ and $\sigma_{\rre}(T)$ denote, respectively, the left essential spectrum and the right essential spectrum of an operator $T$.