A Remark on Hypercontractive Semigroups and Operator Ideals

Details A Remark on Hypercontractive Semigroups and Operator Ideals A Remark on Hypercontractive Semigroups and Operator Ideals

2007-08-25

arxiv.org/abs/0708.3423v2

Mathematics

Functional Analysis

Wolfgang Arendt kindly pointed out to me that the main point of this remark is essentially obvious, just by the analyticity of

Scientific paper

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 $10}$ 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).

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