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On Matsaev's conjecture for contractions on noncommutative $L^p$-spaces (1009.1292v7)

Published 7 Sep 2010 in math.OA and math.FA

Abstract: We exhibit large classes of contractions on noncommutative $Lp$-spaces which satisfy the noncommutative analogue of Matsaev's conjecture, introduced by Peller, in 1985. In particular, we prove that every Schur multiplier on a Schatten space $Sp$ induced by a contractive Schur multiplier on $B(\ell2)$ 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 $\ellp$ and $\sigma$ the shift on the Schatten space $Sp$, the norms $\bnorm{P(S)}{\ellp \xra{}\ellp}$ and $\bnorm{P(\sigma)\ot \Id{Sp}}_{Sp(Sp) \xra{}Sp(Sp)}$ can be different for a complex polynomial $P$.

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