A counterexample to a conjecture of Matsaev

We present an effective algorithm for estimating the norm of an operator mapping a low-dimensional ℓpℓp 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=4p=4. Matsaev conjectured that for every contraction T   on LpLp (1