Some algebraic operators and the invariant subspace problem (Q371722)

The author considers for an invertible operator \(A\) the resolvent algebra \(R_A\) defined by \[ R_A = \{ T \in \mathcal L(X) : \sup_{m \geq 0} \| (1+mA) T (1+mA)^{-1} \| < \infty\} \] and the Deddens algebra \(B_A\) defined by \[ B_A = \{ T \in \mathcal L(X) : \sup_{n \geq 0} \| A^n T A^{-n} \| < \infty\}. \] Since these algebras are not necessarily closed, the author considers instead the variant algebras \[ R_A = \{ T \in \mathcal L(X) : \exists M>0 \text{ such that } \sup_{m \geq 0} \| (1+mA) T (1+mA)^{-1} \| \leq M \|T\| \} \] and \[ B_A = \{ T \in \mathcal L(X) : \exists M>0 \text{ such that } \sup_{n \geq 0} \| A^n T A^{-n} \| \leq M \|T\|\}. \] The goal of the paper under review is to prove the existence of invariant subspaces for \(R_A\) and for \(B_{I-A}\) when \(A\) is an algebraic operator of degree 2 (that is, there is a complex scalar \(\alpha \in \mathbb C\) such that \(A^2 = \alpha A)\). The result for the Deddens algebra requires the additional assumption that \(|\alpha| < |1-\alpha|\). The author also shows that, if \(A\) is an algebraic operator of degree 2, then \[ R_A = \{ S \in \mathcal L(X) : \ker A \in \text{Lat}(S)\}, \] and as a consequence, in the Hilbertian case, if \(\mathcal A \subseteq \mathcal L(H)\) is a non-transitive family of operators, then there is an algebraic operator \(P\) of degree 2 such that \(\mathcal A \subseteq R_P\) and \(\mathcal A \subseteq B_{P-I}\). Finally, the author provides an example where \(R_A\) contains \(\{A\}^\prime\) properly.