Essential spectra of upper triangular relation matrices

Let H and K be infinite dimensional complex separable Hilbert spaces. Given the operators A∈LR(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\in \mathcal{L}\mathcal{R}(H)$$\end{document}, B∈LR(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B\in \mathcal{L}\mathcal{R}(K)$$\end{document} and C∈LR(K,H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C\in \mathcal{L}\mathcal{R}(K,H)$$\end{document}, we define upper triangular linear relation matrix MC:=AC0B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{C}:=\left( {\begin{matrix} A &{}C \\ 0 &{} B \\ \end{matrix}} \right) $$\end{document}. In this paper, we obtain σ⋆(MC)⊆σ⋆(AC(0)H)∪σ⋆(B),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{\star }(M_{C})\subseteq \sigma _{\star }(\left[ {\begin{matrix} A &{}C(0)\\ \end{matrix}} \right] _{H})\cup \sigma _{\star }(B),$$\end{document} and the relations between σ⋆(MC)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{\star }(M_{C})$$\end{document} and σ⋆(A)∪σ⋆(B)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{\star }(A)\cup \sigma _{\star }(B)$$\end{document} are also presented, where σ⋆\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{\star }$$\end{document} is chosen from the essential spectrum, the Weyl spectrum, the essential approximate point spectrum, the Browder spectrum and the Browder essential approximate point spectrum.