von Neumann’s inequality for commuting, diagonalizable contractions. I

Abstract:

We obtain a sufficient condition for an $n$-tuple $T$ of commuting, diagonalizable contractions on a finite-dimensional space to satisfy von Neumann’s inequality $||p(T)|| \leqslant ||p|{|_\infty }$ for any polynomial $p$ in $n$ variables, where $||p|{|_\infty }$ denotes the supremum of $|p|$ over the unit polydisk in ${{\mathbf {C}}^n}$. We apply this condition to the case where $T$ acts on a two- or three-dimensional space. In addition, we prove that von Neumann’s inequality for commuting, diagonalizable contractions on a three-dimensional space implies von Neumann’s inequality for arbitrary commuting contractions on a three-dimensional space.