This is an announcement for the paper "A remark on hypercontractive semigroups and operator ideals" by Gilles Pisier.
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 $\overline{\Gamma_2}$) 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).
Archive classification: math.FA
Mathematics Subject Classification: 47D06
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Submitted from: gip@ccr.jussieu.fr
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