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

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 $P$.