Isometric dilation and Sarason's commutant lifting theorem in several variables

TL;DR

This paper extends isometric dilation and Sarason's commutant lifting theorem to certain classes of commuting contraction tuples in multiple variables, providing explicit dilations and new classes with these properties.

Contribution

It introduces explicit dilations of n-tuples of commuting contractions to BCL-type isometries and establishes a Sarason-type commutant lifting theorem in several variables.

Findings

Operator tuples dilate to BCL-type commuting isometries

Von Neumann inequality holds on a one-dimensional variety

New classes of tuples possess isometric dilations

Abstract

The article deals with isometric dilation and commutant lifting for a class of $n$-tuples $(n \geq 3)$ of commuting contractions. We show that operator tuples in the class dilate to tuples of commuting isometries of BCL type. As a consequence of such an explicit dilation, we show that their von Neumann inequality holds on a one dimensional variety of the closed unit polydisc. On the basis of such a dilation, we prove a commutant lifting theorem of Sarason's type by establishing that every commutant can be lifted to the dilation space in a commuting and norm preserving manner. This further leads us to find yet another class of $n$-tuples $(n\geq 3)$ of commuting contractions each of which possesses isometric dilation.