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

$P$ .

DOI: http://dx.doi.org/10.7900/jot.2010dec29.1905
Keywords: Matsaev's conjecture, noncommutative

$L^p$ -spaces, shift operator, dilations, Schur multipliers, Fourier multipliers, semigroups

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