A condition for asymptotic finite-dimensionality of an operator semigroup (Q662656)

Let \(X\) be a real or complex Banach space. An operator \(T: X \rightarrow X\) is called a power bounded operator if, for all \(n \in \mathbb{N}\), \(\|T^{n}\|\leq C < \infty\), for some positive real number \(C\). Let \(X_{0} = \{x\in X : T^{n}x\rightarrow 0\}\). A power bounded operator on \(X\) is called asymptotically finite-dimensional if \(\text{codim} X_{0} < \infty\). In this article, the author proves that, if there is a compact subset \(K\) of \(X\) such that, for all \(x \in B_{X}\) (the unit ball of \(X\)), \(\liminf \rho(T^{n}x,K)\leq\eta <1\), then for \(\eta < \frac{1}{2}\), \(T\) is asymptotically finite-dimensional. He also shows that for \(X\) reflexive, this result remains valid when \(\frac{1}{2}\leq \eta <1\), but he provides an example to show that the result does not remain true in general. This actually gives a complete answer to a question asked by \textit{E. Yu. Emel'yanov} in his book [Non-spectral asymptotic analysis of one-parameter operator semigroups. Operator Theory: Advances and Applications 173. Basel: Birkhäuser (2007; Zbl 1117.47001)] about the condition for \(T\) to be asymptotically finite-dimensional.