The von Neumann inequality for \(3\times 3\) matrices (Q2788652)

Recall that a \(d\)-tuple of pairwise commuting contractive matrices or operators \(T=(T_1,\dots,T_d)\) satisfies the von Neumann inequality if NEWLINE\[NEWLINE \|p(T)\|\leq\sup_{z\in\mathbb T^d}|p(z)|,\quad p\in\mathbb C[z_1,\dots,z_d], NEWLINE\]NEWLINE where \(\mathbb T^d\) is the unit \(d\)-torus in \(\mathbb C^d\). It is a classical result that the von Neumann inequality holds for \(d\leq2\). For \(d>2\) there are known examples of \(d\)-tuples of commuting contractive finite matrices for which the von Neumann inequality fails. It turns out that the von Neumann inequality holds for \(d\)-tuples of \(2\times2\) commuting contractive matrices. On the other hand, Holbrook has found a \(3\)-tuple of \(4\times4\) matrices for which the von Neumann inequality fails.NEWLINENEWLINEA great deal of effort has been spent to answer what happens when a \(d\)-tuple of \(3\times3\) commuting contracticve matrices is considered.NEWLINENEWLINEThe paper under review explains how \textit{Ł. Kosiński}'s results from [Proc. Lond. Math. Soc. (3) 111, No. 4, 887--910 (2015; Zbl 1327.32022)] provide a proof of the von Neumann inequality for \(d\)-tuples of \(3\times3\) commuting contractive matrices.