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

Abstract

The article deals with isometric dilation and commutant lifting for a class of $n$-tuples $(n\ge 3)$ 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 $(n\ge 3)$ of commuting contractions each of which possesses isometric dilation.

Revised version, Journal reference: Canadian Journal of Mathematics (CJM) (to appear)