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Journal of Operator Theory

Volume 69, Issue 2, Spring 2013  pp. 387-421.

On Matsaev's conjecture for contractions on noncommutative $L^p$-spaces

Authors Cedric Arhancet
Author institution: Laboratoire de Mathematiques, Universite de Franche-Comte, 25030 Besancon Cedex, France

Summary:  We exhibit large classes of contractions on noncommutative $L^p$-spaces which satisfy the noncommutative analogue of Matsaev's conjecture, introduced by Peller. In particular, we prove that every Schur multiplier on a Schatten space $S^p$ induced by a contractive Schur multiplier on $B(\ell^2)$ associated with a real matrix satisfy this conjecture. Moreover, we deal with analogue questions for $C_0$-semigroups. Finally, we disprove a conjecture of Peller concerning norms on the space of complex polynomials arising from Matsaev's conjecture and Peller's problem. Indeed, if $S$ denotes the shift on $\ell^p$ and $\sigma$ the shift on the Schatten space $S^p$, the norms $\|P(S)\|_{\ell^p \xrightarrow\ell^p}$ and $\|P(\sigma)\ot \Id_{S^p}\|_{S^p(S^p) \xrightarrow S^p(S^p)}$ can be different for a complex polynomial $P$.

Keywords:  Matsaev's conjecture, noncommutative $L^p$-spaces, shift operator, dilations, Schur multipliers, Fourier multipliers, semigroups

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