A Remark on Hypercontractive Semigroups and Operator Ideals

In this note, we answer a question raised by Johnson and Schechtman \cite{JS}, about the hypercontractive semigroup on $\{-1,1{\}}^{\text{NN}}$. 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 $\overline{{\mathrm{\Gamma }}_2}$) of the subset (denoted by ${\mathrm{\Gamma }}_2$) formed by the operators mapping $L_p$ to $L_2$ (a fortiori these operators factor through a Hilbert space).