Algebra descent spectrum of operators
The descent \(d(T)\) of a linear operator \(T\) acting on a vector space \(X\) over a field \(K\) is defined by \(d(T):= \min\{n \in \mathbb{N}\cup\{0\}: R(T^n)=R(T^{n+1})\}\) with the convention that \(\min \emptyset = \infty\). Here, \(R(\cdot)\) denotes the range. The descent spectrum of \(T\) is the set \(\sigma_{\text{desc}}(T):= \{\lambda\in K: d(T -\lambda) = \infty\}\). Clearly, \(\sigma_{\text{desc}}(T)\) is a subset of the usual spectrum \(\sigma(T):=\{\lambda \in K : T -\lambda \text{~is~not~bijective} \}\). Also, \(\sigma_{\text{desc}}(T,B(X)) := \sigma_{\text{desc}}(TM_T)\), with \(M_T\) the multiplication operator \(M_T(S) = TS\) on \(B(X)\). The ``descent spectrum equality'' (in short, DSE) means that for every \(T\) in the algebra \(B(X)\) of all bounded linear operators on \(X\), the equality \(\sigma_{\text{desc}}(T)=\sigma_{\text{desc}}(T, B(X))\) holds. The authors investigate the interesting question ``Does every Banach space \(X\) satisfy the DSE?''. They prove that the DSE is fulfilled by \(\ell_1(I)\) and \(\ell_2(I)\) for any set \(I\), but not by \(\ell_p(I)\) for any infinite set \(I\) and any \(p\neq 2\) with \(1 < p \leq \infty\). They show that the DSE is also fulfilled by all Banach spaces which are not isomorphic to any of their proper quotients. Actually, a Banach space is not isomorphic to any of its proper quotients if and only if it is not isomorphic to any of its proper complemented subspaces and satisfies the DSE. Another characterization of Banach spaces which are not isomorphic to any of their proper quotients is the one asserting that they are precisely those Banach spaces \(X\) such that, for every \(T \in B(X)\), the interior of \(\sigma(T)\) is contained in \(\sigma_{\text{desc}}(T )\).
scientific article; zbMATH DE number 5826053
scientific article