Some pseudo-spectrum equalities of 2 × 2 unbounded upper triangular operator matrices and applications

Abstract

Let $\epsilon >0$,$T=\left(\begin{array}{cc}A& B\\ 0& D\end{array}\right):D\left(A\right)\times D\left(D\right)\subset X\times X\to X\times X.$ We investigate the conditions under which${\sigma }_{⁎}\left(T\right)={\sigma }_{⁎}\left(A\right)\cup {\sigma }_{⁎}\left(D\right)({\sigma }_{⁎}={\sigma }_{\epsilon },{\sigma }_{ap,\epsilon },{\sigma }_{\delta ,\epsilon },{\sigma }_{e5,\epsilon },{\sigma }_{eap,\epsilon },{\sigma }_{e\delta ,\epsilon })$ hold and some sufficient conditions are obtained, where the set ${\sigma }_{\epsilon }(\cdot )$ denotes the pseudo-spectrum, ${\sigma }_{ap,\epsilon }(\cdot )$ and ${\sigma }_{\delta ,\epsilon }(\cdot )$ denote the approximation pseudo-spectrum and defect pseudo-spectrum, ${\sigma }_{e5,\epsilon }(\cdot )$ denotes the Ammar-Jeribi essential pseudo-spectrum, ${\sigma }_{eap,\epsilon }(\cdot )$ and ${\sigma }_{e\delta ,\epsilon }(\cdot )$ denote the essential approximation pseudo-spectrum and essential defect pseudo-spectrum, respectively. In the end, the main results are applied to the boundary value problem of the plate bending equation, and it is verified that the obtained conclusion is consistent with the result of direct calculation.