Slowly Changing Vectors and the Asymptotic Finite-Dimensionality of an Operator Semigroup

TL;DR

This paper investigates the structure of linear power bounded operators on Banach spaces, showing conditions under which the space decomposes into finite-dimensional parts based on the asymptotic behavior of the operator.

Contribution

It introduces new techniques to relate the asymptotic properties of operators and semigroups to the finite-dimensionality of certain subspaces in Banach spaces.

Findings

Existence of eigenvalues related to vectors not tending to zero.

Finite codimension of the subspace of vectors tending to zero under certain conditions.

Results extend to one-parameter semigroups.

Abstract

Let $T:X\to X$ be a linear power bounded operator on Banach space. Let $X_0$ is a subspace of vectors tending to zero under iterating of $T$. We prove that if $X_0$ is not equal to $X$ then there exists $\lambda$ in Sp(T) such that, for every $\epsilon>0$, there is $x$ such that $|Tx-\lambda x|<\epsilon $ but $|T^nx|>1-\epsilon$ for all $n$. The technique we develop enables us to establish that if $X$ is reflexive and there exists a compactum $K$ in $X$ such that for every norm-one $x\in X$ $\rho\{T^nx, K\}<\alpha (T)<1$ for some $n=n_1, n_2,...$ then $codim(X_0)<\infty$. The results hold also for a one-parameter semigroup.