A note on the periodic decomposition problem for semigroups

TL;DR

This paper investigates conditions under which the kernel of a product of commuting power-bounded operators equals the sum of their kernels, focusing on semigroup-generated operators and generalizations of the periodic decomposition problem.

Contribution

It extends the periodic decomposition problem to operators generated by semigroups and explores conditions for kernel equality in this context.

Findings

Identifies conditions for kernel equality in semigroup-generated operators.

Generalizes the periodic decomposition problem beyond cyclic semigroups.

Provides insights into the structure of kernels for commuting power-bounded operators.

Abstract

Given $T_1,\dots, T_n$ commuting power-bounded operators on a Banach space we study under which conditions the equality $\ker (T_1-\mathrm{I})\cdots (T_n-\mathrm{I})=\ker(T_1-\mathrm{I})+\cdots +\ker (T_n-\mathrm{I})$ holds true. This problem, known as the periodic decomposition problem, goes back to I. Z. Ruzsa. In this short note we consider the case when $T_j=T(t_j)$, $t_j>0$, $j=1,\dots, n$ for some one-parameter semigroup $(T(t))_{t\geq 0}$. We also look at a generalization of the periodic decomposition problem when instead of the cyclic semigroups $\{T_j^n:n \in \mathbb{N}\}$ more general semigroups of bounded linear operators are considered.