On the reflexivity of multivariable isometries

Abstract

Let A\subsetC(K) be a unital closed subalgebra of the algebra of all continuous functions on a compact set K in \mathbb{C}^{n}. We define the notion of an A-isometry and show that, under a suitable regularity condition needed to apply Aleksandrov's work on the inner function problem, every A-isometry T\in L(\mathcal{H})^{n} is reflexive. This result applies to commuting isometries, spherical isometries, and more general, to all subnormal tuples with normal spectrum contained in the Bergman-Shilov boundary of a strictly pseudoconvex or bounded symmetric domain.