Ces\`aro-Hardy operators on $L^p[0,1]$: fine spectrum, weighted Koopman semigroups and invariant subspaces

TL;DR

This paper analyzes the spectral properties and invariant subspaces of Cesàro-Hardy operators on L^p[0,1], utilizing semigroup theory and functional calculus to reveal their boundedness and spectral characteristics.

Contribution

It introduces a novel approach to study Cesàro-Hardy operators via $C_0$-semigroups, connecting their spectra to the generators of these semigroups and exploring invariant subspace implications.

Findings

Spectral properties of Cesàro-Hardy operators are characterized using semigroup generators.

The study demonstrates the universality of certain translations related to the semigroup $T(t)$.

Results on invariant subspaces of Cesàro-Hardy operators are established.

Abstract

In this paper we study boundedness and detailed spectral properties for the Ces\`aro-Hardy operator and some generalizations in $L^p[0,1]$. The study employs $C_0$-semigroup theory, expressing the Ces\`aro-Hardy operators and their dual operators through subordination with $C_0$-semigroups $T(t)$ and $S(t)$ respectively. The spectral properties of the semigroup's infinitesimal generators are transferred to the Ces\`aro-Hardy operators using functional calculus methods. Furthermore, some implications for the Invariant Subspace Problem are explored by demonstrating the universality of certain translations related to the semigroup $T(t)$, and providing results on the invariant subspaces of these operators.