ON MATSAEV'S CONJECTURE FOR CONTRACTIONS ON NONCOMMUTATIVE Lp-SPACES

We exhibit large classes of contractions on noncommutative L-p-spaces which satisfy the noncommutative analogue of Matsaev's conjecture, introduced by Feller. In particular, we prove that every Schur multiplier on a Schatten space S-p induced by a contractive Schur multiplier on B(l(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 Feller concerning norms on the space of complex polynomials arising from Matsaev's conjecture and Peller's problem. Indeed, if S denotes the shift on l(p) and sigma the shift on the Schatten space S-p, the norms parallel to P(S)parallel to(lp -> lp) and parallel to P(sigma) circle times Id(Sp) parallel to(Sp(Sp) -> Sp(Sp)) can be different for a complex polynomial P.