Abstract of a paper by Gilles Pisier
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 The source file(s), hyper.tex: 11355 bytes, is(are) stored in gzipped form as 0708.3423.gz with size 5kb. The corresponding postcript file has gzipped size 50kb. Submitted from: gip@ccr.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0708.3423 or http://arXiv.org/abs/0708.3423 or by email in unzipped form by transmitting an empty message with subject line uget 0708.3423 or in gzipped form by using subject line get 0708.3423 to: math@arXiv.org.
participants (1)
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Dale Alspach