On operators with closed analytic core (Q2569789)

The analytic core \(K(T)\) of a bounded operator \(T\) on the Banach space \(X\) is the set of all \(x\in X\) for which there exist a constant \(c>0\) and a sequence of elements \(x_n\in X\) such that \(x_0=x\), \(Tx_n=x_{n-1}\) and \(\|x_n\|\leq c^n\|x\|\) for all \(n\in\mathbb N\). Since clearly \(K(T)=X\) if \(T\) is invertible, the interesting case is when \(T\) is not invertible. It is known that if \(0\) is an isolated point of \(\sigma(T)\), the spectrum of \(T\), then \(K(T)\) is closed. In the present paper, the authors use techniques from local spectral theory to find conditions under which the converse is true. One result is that for decomposable non-invertible operators, \(0\) is an isolated point of the spectrum if and only in the analytic core is closed. Another result is that if \(T\) is a Riesz operator, then \(K(T)\) is closed if and only if \(\sigma(T)\) is finite.