A counterexample to a conjecture of Matsaev

We present an effective algorithm for estimating the norm of an operator mapping a low-dimensional ?^p space to a Banach space with an easily computable norm. We use that algorithm to show that Matsaev’s proposed extension of the inequality of John von Neumann is false in case p=4. Matsaev conjectured that for every contraction T on L^p (1<p<∞) one has for any polynomial P

‖P(T)‖_L^p→L^p?‖P(S)‖_{?^p(Z⁺)→?^p(Z⁺)}