The point spectrum and residual spectrum of upper triangular operator matrices

Abstract

The point and residual spectra of an operator are, respectively, split into 1,2-point spectrum and 1,2-residual spectrum, based on the denseness and closedness of its range. Let $\mathcal{H}$, $\mathcal{K}$ be infinite dimensional complex separable Hilbert spaces and write $M_{X}=\left(\begin{smallmatrix} A & X\\ 0 & B \end{smallmatrix}\right)\in\mathcal{B}(\mathcal{H}\oplus\mathcal{K})$. For given operators %the diagonal entries $A\in \mathcal{B}(\mathcal{H})$ and $B\in\mathcal{B} (\mathcal{K})$, the sets $ \bigcup\limits_{X\in\mathcal{B}(\mathcal{K},\mathcal{H})} \sigma_{*,i}(M_{X})$ $(*=p,r; i=1,2)$, are characterized. Moreover,  we obtain some necessary and sufficient condition such that $\sigma_{*,i}(M_{X})= \sigma_{*,i}(A)\cup\sigma_{*,i}(B)$ $(*=p,r; i=1,2)$ for every $X\in\mathcal{B}(\mathcal{K},\mathcal{H})$.