The essential spectrum of AM-compact operators
The paper is devoted to the construction of a representation space for the essential spectrum of an operator in the category of Banach lattices. If \(E\) is a Banach lattice, \(s(E)\) denotes the space of those sequences \((x_ n)\) in \(E\) such that its range \(\{x_ n:\;n\in N\}\) is almost order bounded. The quotient space \(\ell_{\infty}(E)/s(E)=\tilde E\) is a Banach lattice, too. An operator \(T: E\to E\) is called quasi regular if \(T\) maps order bounded sets into almost order bounded sets. The authors show that \({\mathcal L}^{qr}=\{\) quasi-regular operators from \(E\) into \(E\}\) is a closed subalgebra of \({\mathcal L}(E)\) and that any \(T\in{\mathcal L}^{qr}(E)\) induces an operator \(\tilde T\) in \({\mathcal L}(\tilde E)\) by the formula \(\tilde T[(x_ n)+s(E)]:=(Tx_ n)+s(E)\). Then they consider \(AM\)-compact operators (i.e. operators mapping order bounded sets onto relatively compact sets). Since each \(AM\)-compact operator \(T\) is quasi- regular, \(T\) induces a \(\tilde T\in{\mathcal L}(\tilde E)\); furthermore \(T(s(E))\subset m(E)=\{(x_ n)\subset E\) that have a relatively compact range\(\}\). This inclusion is basic to show the following result: Theorem 1.5. Let \(E\) be a Banach lattice and \(T\in{\mathcal L}(E)\) be an \(AM\)-compact operator. Then (i) \(\sigma_ p(\tilde T)-\{0\}=\sigma_ p(P(T))-\{0\}\), (ii) if \(\lambda\neq 0\), then \(\lambda I-\tilde T\) is surjective if and only if \(\lambda I-P(T)\) is surjective, (iii) \(\sigma(\tilde T)-\{0\}=\sigma(P(T))-\{0\}\), (iv) \(\sigma^ +_{ess}(T)-\{0\}=\sigma_ p(\tilde T)-\{0\}\), \(\sigma^ -_{ess}(T)-\{0\}=\{\lambda\neq 0:\;\lambda I-\tilde T\text{ is not surjective}\}\) and \(\sigma_{ess}(T)-\{0\}=\sigma(\tilde T)- \{0\}\) (where \(\sigma(T)\) denotes the spectrum of \(T\), \(\sigma_ p(T)\) its point spectrum, \(\sigma_{ess}(T)\) the essential spectrum of \(T\), \(P(T)\) the bounded linear operator induced from \(\ell_ \infty(E)/m(E)\) to itself by \(T\in{\mathcal L}(E)\) and defined by \(P(T)[(x_ n)+m(E)]:=(Tx_ n)+m(E)\) for \((x_ n)\in\ell_ \infty(E))\). Then they use the above result to get several consequences: among them the following ones: Corollary 1.7. Let \(E\) be a Banach lattice and \(0\leq T\in{\mathcal L}(E)\) be an \(AM\)-compact operator. Then the following statements hold: (i) If \(T\) is a lattice homomorphism, then \(\sigma_{ess}(T)\) is cyclic, (ii) If one of the following conditions (a) \(\{(\lambda-r_{ess}(T))R(\lambda,T):\;\lambda>r_{ess}(T)\}\) is bounded in \({\mathcal L}(\tilde E)\), (b) \(r_{ess}(T)=1\) and \(\| T^ n\|\leq M\) for all \(n\in N\), holds, then the peripherical essential spectrum \(\pi\sigma_{ess}(T):=\{\lambda\in\sigma_{ess}(T):\;|\lambda|=1\}\) is cyclic. Corollary 1.9. Let \(E\) be a Banach lattice. Then the following statements hold: (i) \(r_{ess}(T)\in\sigma_{ess}(T)\) for any positive \(AM\)-compact operator \(T\) in \(E\), (ii) Suppose that \(S,T\in{\mathcal L}(E)\) are such that \(0\leq S\leq T\) and \(S\) is \(AM\)-compact. Then \(r_{ess}(S)\leq r_{ess}(T)\). In the second part of the note the authors consider (essentially) semicompact operators, i.e. operators mapping norm bounded (sequences) sets into almost order bounded (sequences) sets. Then they show Proposition 2.3. Let \(E\) be a Banach lattice. Then \({\mathcal S}_ s(E)\) and \({\mathcal S}(E)\) are closed ideals of \({\mathcal L}^{qr}(E)\) (where \({\mathcal S}(E):=\{T\in{\mathcal L}(E):\;T\text{ is semicompact}\}\), \({\mathcal S}_ s(E):=\{T\in{\mathcal L}(E):\;T\) is sequentially semicompact\(\}\)). Proposition 2.4. Let \(E\) be a Banach lattice and \(T\in{\mathcal L}^{qr}(E)\). Then the following hold: (i) \(\sigma(\tilde T)\subseteq\sigma_ s(T)\). (ii) If \(T\) is \(AM\)-compact, then \(\sigma_{ess}(T)-\{0\}=\sigma(\tilde T)-\{0\}=\sigma_ s(T)-\{0\}\). At the end they consider the measure of non-semicompactness introduced by \textit{B. de Pagter} and \textit{A. R. Schep} [J. Funct. Anal. 78, 31-55 (1988; Zbl 0651.47023)] and prove the following final result: Proposition 2.5. Let \(E\) be a Banach lattice. Then: (i) \(\|(x_ n)+s(E)\|=\rho(\{x_ n:\;n\in N\})\) for any \((x_ n)\in\ell_ \infty(E)\). (ii)\(\|\tilde T\|\leq\rho(T)\) for \(T\in{\mathcal L}^{qr}(E)\), with equality if \(E\) is separable. (iii) \(\rho(T)\leq\| T\|_ s\) for \(T\in{\mathcal L}^{qr}(E)\) if \(E\) has order continuous norm or a quasi-interior point.
scientific article; zbMATH DE number 11706
scientific article