Von Neumann’s inequality for commuting operator-valued multishifts

Abstract

Recently, Hartz proved that every commuting contractive classical multishift with non-zero weights satisfies the matrix-version of von Neumann's inequality. We show that this result does not extend to the class of commuting operator-valued multishifts with invertible operator weights. In fact, we show that if A and B are commuting contractive d-tuples of operators such that B satisfies the matrix-version of von Neumann's inequality and (1,..., 1) is in the algebraic spectrum of B, then the tensor product A circle times B satisfies von Neumann's inequality if and only if A satisfies von Neumann's inequality. We also exhibit several families of operator-valued multishifts for which von Neumann's inequality always holds.