Eventual Ideal Properties of the Riemann-Liouville Analytic Semigroup

TL;DR

This work systematically analyzes the Riemann–Liouville analytic semigroup $\{V_\xi\}_{\xi\in\mathbb{C}_0}$ on spaces $L^p(0,1)$ and $C([0,1])$, establishing sharp, threshold-based membership criteria in major operator ideals. Central results include: a complete Schatten-class characterization on $L^2(0,1)$ with $V_\xi\in\mathcal{S}^r$ iff $\text{Re}(\xi)>1/r$; a precise description of when $V_\xi$ is nuclear or absolutely $p$-summing on $L^p(0,1)$, with explicit region constraints on $\xi$ depending on $p$ and $r$; and parallel results for $C([0,1])$ involving $r$-integral and absolutely $r$-summing properties. The paper also proves unicellularity of $V_\xi$ and details the invariant-subspace structure, providing a thorough operator-theoretic portrait of the RL semigroup. Together, these findings illuminate how fractional integration semigroups interact with classical operator ideals, with consequences for spectral theory, compactness, and summability properties in harmonic-analytic settings.

Abstract

In this paper, we revisit the Riemann--Liouville analytic semigroup. In particular, we completely characterize the membership to the Schatten class $S^r$ on $L^2(0,1)$, as well as the membership to the class of nuclear operators on $L^p(0,1)$, $p\geq 1$, and the membership to the ideal of absolutely $r$-summing operators for any $r\geq 1$.

Theorems & Definitions (57)

• Theorem 2.1: Pietsch
• Theorem 2.2: Grothendieck
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Theorem 2.5
• proof
• Theorem 2.6: Salem
• proof