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

TL;DR

This paper investigates classes of contractions on noncommutative L^p-spaces satisfying a noncommutative version of Matsaev's conjecture, providing new results for Schur multipliers and disproving related conjectures by Peller.

Contribution

It identifies broad classes of contractions satisfying the noncommutative Matsaev's conjecture and disproves a conjecture regarding polynomial norms in this context.

Findings

Schur multipliers induced by real matrices satisfy the conjecture

Disproved Peller's conjecture on polynomial norms in noncommutative L^p-spaces

Established results for C_0-semigroups related to the conjecture

Abstract

We exhibit large classes of contractions on noncommutative $L^p$-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 $S^p$ 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 $\bnorm{P(S)}_{\ell^p \xra{}\ell^p}$ and $\bnorm{P(\sigma)\ot \Id_{S^p}}_{S^p(S^p) \xra{}S^p(S^p)}$ can be different for a complex polynomial $P$.