On von Neumann's inequality for complex triangular Toeplitz contractions (Q2181176)

Let \(n\) be a natural number, and let \(T = (T_1, \dots, T_n)\) be an \(n\)-tuple of commuting contractions acting on some Hilbert space \(\mathcal{H}\) (over \(\mathbb{C}\)). We say that \(T\) satisfies the von Neumann inequality if \[ \|p(T_1, \dots, T_n)\|_{\mathcal{B}(\mathcal{H})} \leq \sup \{|p(z_1, \dots, z_n)|: |z_1|, \dots, |z_n| <1\}. \] If \(n=1, 2\), then the above inequality holds for all \(T\) and all \(\mathcal{H}\). However, this is not true if \(n > 2\). The inequality fails even if we restrict to \(3\) tuples of commuting contractions on finite dimensional Hilbert spaces (so long as the dimension of the Hilbert space is larger than \(3\)). The author proves, among other things, that the von Neumann inequality holds for all \(n\)-tuples of commuting circulant matrices.