Self-adjoint perturbations of left (right) Weyl spectrum for upper triangular operator matrices

Abstract

Let $\mathcal{H}$ be a separable infinite-dimensional Hilbert space.
Given the operators $A\in\mathcal{B}(\mathcal{H})$ and $B\in\mathcal{B}(\mathcal{H}),$
we define $M_{X}:= \begin{bmatrix}\begin{smallmatrix}
A& X\\
0& B
\end{smallmatrix}\end{bmatrix}$ where $X\in \mathcal{S}(\mathcal{H})$ is a self-adjoint operator.
In this paper, a necessary and sufficient condition is given for $M_{X}$ to be a left (right) Weyl operator for some $X\in\mathcal{S}(\mathcal{H})$.
Moreover, it is shown that
\[\begin{array}{l}
\bigcap\limits_{X\in\mathcal{S}(\mathcal{H})} \sigma_{\star}(M_{X})=\bigcap\limits_{X\in\mathcal{S}(\mathcal{H})\cap {\rm Inv} (\mathcal{H})} \sigma_{\star}(M_{X})
=\bigcap\limits_{X\in\mathcal{B}(\mathcal{H})} \sigma_{\star}(M_{X})\cup\Delta,
\end{array}
\]
where $\sigma_{*}$ is the left (right) Weyl spectrum.
Finally, we further characterize the perturbation of the left (right) Weyl spectrum for Hamiltonian operators.