Dynamics of a compact operator

TL;DR

This paper investigates the behavior of iterates and their averages for compact linear or affine operators on Banach spaces, showing they are either bounded or diverge to infinity.

Contribution

It characterizes the dynamics of iterates of compact operators on Banach spaces, extending understanding of their boundedness or divergence behavior.

Findings

Iterates of compact operators are either bounded or diverge to infinity.

Averages of iterates exhibit similar boundedness or divergence.

The results apply to both linear and affine compact operators.

Abstract

Let $T:X\to X$ be a compact linear (or more generally affine) operator from a Banach space into itself. For each $x\in X$, the sequence of iterates $T^nx, n=0,1,...$ and its averages $\frac{1}{k}\sum_{k=0}^nT^{k-1}x, n=0,1,...$ are either bounded or approach infinity.