The essential spectrums of \(2\times 2\) unbounded anti-triangular operator matrices

Let \(\begin{aligned} {\mathcal {T}}=\left( \begin{array}{cc} 0 &{}\quad B \\ C &{}\quad D \\ \end{array} \right) :D(C)\times D(B)\subset X\times X\rightarrow X\times X \end{aligned}\) $$\begin{array}{c}\hfill \mathcal{T}=\left(\begin{array}{cc}0& B\\ C& D\end{array}\right):D(C)\times D(B)\subset X\times X\to X\times X\end{array}$$ be a \(2\times 2\) $2\times 2$ unbounded anti-triangular operator matrix on complex Hilbert space \(X\times X\) $X\times X$ . Using the relative compact perturbation theory and the space decomposition method, the seven essential spectrum equalities are characterized as \(\begin{aligned} \sigma _{ei}({\mathcal {T}})=\{\lambda \in \mathbb C:\lambda ^2\in \sigma _{ei}(BC)\cup \sigma _{ei}(CB)\},~~~~i\in \{1,~2,~3,~4,~5,~6,~7\}, \end{aligned}\) $$\begin{array}{c}\hfill {\sigma }_{\mathit{ei}}(\mathcal{T})=\{\lambda \in \mathbb{C}:{\lambda }^2\in {\sigma }_{\mathit{ei}}(BC)\cup {\sigma }_{\mathit{ei}}(CB)\},i\in \{1,2,3,4,5,6,7\},\end{array}$$ where \(\sigma _{ei}(\cdot )\) ${\sigma }_{\mathit{ei}}(·)$  ( \(i=1,\ldots ,7\) $i=1,\dots ,7$ ) denote the Gustafson essential spectrum, Weidmann essential spectrum, Kato essential spectrum, Wolf essential spectrum, Schechter essential spectrum, essential approximation point spectrum, and essential defect spectrum, respectively. An example is also provided to illustrate the validity of the criterion.