A counterexample to a conjecture of Matsaev (Q2431193)

The conjecture of Matsaev asserts that, for every contraction \(T\) on \(L^p\), \(1<p<\infty\), and for any complex polynomial \(P\), we have \(\|P(T)\|_{L^p\rightarrow L^p} \leq \|P(S)\|_{\ell^p\rightarrow \ell^p}\), where \(S\) is the unilateral shift. For \(p=2\), it is a consequence of von Neumann's inequality. The author of this paper presents an algorithm for estimating the norm of an operator mapping an \(\ell^p\) space to a Banach space with easily computable norm. The algorithm is used to show that, for \(p=4\), Matsaev's conjecture is false.