von Neumann Inequality for a class of Doubly Contractive Weighted Shifts

TL;DR

This paper extends the von Neumann inequality to certain doubly contractive weighted shifts and commuting tuples, showing conditions under which they admit spherical unitary dilations and satisfy the inequality.

Contribution

It establishes the von Neumann inequality for a class of doubly contractive weighted shifts and certain commuting operator tuples, under specific weight conditions.

Findings

Weighted shifts that are balanced admit spherical unitary dilations.

Such tuples satisfy the von Neumann inequality over the Euclidean unit ball.

The inequality is also established for homogeneous polynomials of degree at most 2.

Abstract

In this article, we investigate the ball version of von Neumann inequality for the class of doubly contractive $d$-tuple of weighted shift. We show that if the weighted shift is balanced or satisfies an appropriate weight condition, then it admits a spherical unitary dilation. Consequently, such tuples satisfy the von Neumann inequality over Euclidean unit ball. For the general class of commuting tuple of doubly contractive operators (not necessarily weighted shift) on a Hilbert space, we further establish von Neumann inequality for homogeneous polynomials of degree at most $2.$