A Remark on Hypercontractive Semigroups and Operator Ideals

TL;DR

This paper investigates hypercontractive semigroups on $L_p$ spaces, proving that under certain conditions, these semigroups are in the norm closure of operators factoring through a Hilbert space, answering a question by Johnson and Schechtman.

Contribution

It establishes a general theorem linking hypercontractivity of semigroups on $L_p$ spaces to their approximation by operators factoring through Hilbert spaces, extending prior understanding.

Findings

Semigroups with mild hypercontractivity are in the closure of Hilbert space factorization operators.

The result applies to the hypercontractive semigroup on $ ext{-}1,1 ext{}^{ n}$.

Provides a positive answer to a question by Johnson and Schechtman.

Abstract

In this note, we answer a question raised by Johnson and Schechtman \cite{JS}, about the hypercontractive semigroup on $\{-1,1\}^{\NN}$. More generally, we prove the folllowing theorem. Let $1<p<2$. Let $(T(t))_{t>0}$ be a holomorphic semigroup on $L_p$ (relative to a probability space). Assume the following mild form of hypercontractivity: for some large enough number $s>0$, $T(s)$ is bounded from $L_p$ to $L_2$. Then for any $t>0$, $T(t)$ is in the norm closure in $B(L_p)$ (denoted by $\bar{\Gamma_2}$) of the subset (denoted by ${\Gamma_2}$) formed by the operators mapping $L_p$ to $L_2$ (a fortiori these operators factor through a Hilbert space).