Essential spectra of some matrix operators and application to two-group transport operators with general boundary conditions
Let \(X\) be a Banach space, \({\mathcal L}(X)\) be the set of all bounded linear operators on \(X\), and \({\mathcal C}(X)\) be the set of all closed densely defined linear operators on \(X\). Let \(\Phi_-(X)\) and \(\Phi_+(X)\) be the sets of lower and upper semi-Fredholm operators in \({\mathcal C}(X)\) and \(\Phi(X)=\Phi_-(X)\cap\Phi_+(X)\) be the set of all Fredholm operators on \(X\). The authors study the following essential spectra of \(T\in{\mathcal C}(X)\): the Gustafson--Weidmann essential spectra \(\sigma_{e1}:= \{\lambda\in {\mathbb C}:\lambda-T\notin\Phi_+(X)\}\), \(\sigma_{e2}:=\{\lambda\in {\mathbb C}:\lambda-T\notin\Phi_-(X)\}\); the Kato essential spectrum \(\sigma_{e3}:=\{\lambda\in {\mathbb C}:\lambda-T\notin\Phi_-(X) \cup\Phi_+(X)\}\); the Wolf essential spectrum \(\sigma_{e4}:=\{\lambda\in {\mathbb C}:\lambda-T\notin\Phi(X)\}\); the Schechter essential spectrum \(\sigma_{e5}(T):= {\mathbb C}\setminus\rho_5(T)\), where \(\rho_5(T):=\{\lambda\in{\mathbb C}: \lambda-T\in\Phi(X),\;\text{ind}(\lambda-T)=0\}\); and the Browder essential spectrum \(\sigma_{e6}(T):= {\mathbb C}\setminus\rho_6(T)\), where \(\rho_6(T)\) denotes the set of those \(\lambda\in\rho_5(T)\) such that all scalars near \(\lambda\) belong to the resolvent \(\rho(T)\). An operator \(F\in{\mathcal L}(X)\) is called a Fredholm perturbation (resp., an upper semi-Fredholm perturbation, a lower semi-Fredholm perturbation) if \(U+F\in\Phi(X)\) (resp., \(U+F\in\Phi_+(X)\), \(U+F\in\Phi_-(X)\)) whenever \(U\in\Phi(X)\) (resp., \(U\in\Phi_+(X)\), \(U\in\Phi_-(X)\)). The set of all Fredholm (lower semi-Fredholm, upper semi-Fredholm) perturbations is denoted by \({\mathcal F}(X)\) (resp., \({\mathcal F}_-(X)\), \({\mathcal F}_+(X)\)). Suppose that \(L_0= \left(\begin{matrix} A & B \\ C & D \end{matrix}\right)\) is defined on \(X\times X\), where \(A\) is closed densely defined linear operator on \(X\) with nonempty resolvent set \(\rho(A)\); \(B\) is densely defined linear operator on \(X\) and, for some \(\mu\in\rho(A)\), the operator \((A-\mu)^{-1}B\) is closable; \({\mathcal D}(A)\subset{\mathcal D}(C)\) and, for some \(\mu\in\rho(A)\), the operator \(C(A-\mu)^{-1}\) is bounded; \({\mathcal D}(B)\cap{\mathcal D}(D)\) is dense in \(X\) and, for some \(\mu\in\rho(A)\), the operator \(D-C(A-\mu)^{-1}B\) is closable; in that case, its closure is denoted by \(S(\mu)\). \textit{F.~V.\ Atkinson, H.~Langer, R.~Mennicken} and \textit{A.~A.\ Shkalikov} [Math.\ Nachr.\ 167, 5--20 (1994; Zbl 0831.47001)] proved that under these assumptions \(L_0\) is closable and gave an explicit formula for its closure \(L\). Set \[ M(\mu):=\left(\begin{matrix} 0 & \overline{(\mu-A)^{-1}B}\\ C(\mu-A)^{-1} & C(\mu-A)^{-1} \overline{(\mu-A)^{-1}B} \end{matrix}\right). \] The main result of the paper under review is the following. Theorem. If \({\mathcal I}(X)\) is a nontrivial ideal of \({\mathcal L}(X)\), which is contained in \({\mathcal F}(X)\), and \(C(A-\mu)^{-1}\in{\mathcal I}(X)\) for some \(\mu\in\rho(A)\), then the following statements hold. (i) If \(M(\mu)\in{\mathcal F}(X\times X)\) for some \(\mu\in\rho(A)\), then \(\sigma_{e4}(L)=\sigma_{e4}(A) \cup\sigma_{e4}(S(\mu))\) and \(\sigma_{e5}(L)\subset\sigma_{e5}(A) \cup\sigma_{e5}(S(\mu))\). Moreover, if \({\mathbb C}\setminus\sigma_{e4}(A)\) is connected, then \(\sigma_{e5}(L)=\sigma_{e5}(A) \cup\sigma_{e5}(S(\mu))\). If, in addition, \({\mathbb C}\setminus\sigma_{e5}(L)\) is connected, \(\rho(L)\neq\emptyset\), \({\mathbb C} \setminus\sigma_{e5}(S(\mu))\) is connected, and \(\rho(S(\mu))\neq\emptyset\), then \(\sigma_{e6}(L)=\sigma_{e6}(A) \cup\sigma_{e6}(S(\mu))\). (ii) If \({\mathcal I}(X)\subset{\mathcal F}_+(X)\) and \(M(\mu)\in{\mathcal F}_+(X\times X)\) for some \(\mu\in\rho(A)\), then \(\sigma_{e1}(L)=\sigma_{e1}(A) \cup\sigma_{e1}(S(\mu))\). (iii) If \({\mathcal I}(X)\subset{\mathcal F}_-(X)\) and \(M(\mu)\in{\mathcal F}_-(X\times X)\) for some \(\mu\in\rho(A)\), then \(\sigma_{e2}(L)=\sigma_{e2}(A) \cup\sigma_{e2}(S(\mu))\). (iv) If \({\mathcal I}(X)\subset{\mathcal F}_+(X)\cap{\mathcal F}_-(X)\) and \(M(\mu)\in{\mathcal F}_+(X\times X)\cap{\mathcal F}_-(X\times X)\) for some \(\mu\in\rho(A)\), then \[ \sigma_{e3}(L)=\sigma_{e3}(A) \cup\sigma_{e3}(S(\mu))\cup \big[\sigma_{e2}(A) \cap\sigma_{e1}(S(\mu))\big] \cup \big[\sigma_{e1}(A) \cap\sigma_{e2}(S(\mu))\big]. \] This result is applied in the study of the essential spectra of two-group transport operators with general boundary conditions in \(L_p([-a,a]\times[-1,1])\times L_p([-a,a]\times[-1,1])\), where \(a>0\).
scientific article; zbMATH DE number 5077890
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