2026/07/15 by Kelly Bickel, J. E. Pascoe, Ryan Tully-Doyle
#math.FA #math.CA
Given an n × n strictly contractive matrix T, an (automorphic) Nelson dilation \widehatT of T is a certain type of analytic matrix-valued function on the unit disk with \widehatT(0) = T. Its construction gives a method for lifting a matrix to a matrix-valued function with nice boundary behavior, a trick that has proved useful in recent operator theoretic developments. In this paper, we show that Nelson dilations give a quick way to obtain the minimal isometric and unitary dilations of T and thus, connect naturally to the classical Sz.-Nagy dilation theory. We then initiate the study of the automorphic Nelson dilations as a fundamental object in their own right and prove that every T has Nelson dilations \widehatT with particularly useful/interesting properties; for example, they either have strongly entangled eigenvalue functions or have reducing subspaces that are independent of z. Along the way, we examine when the product of an invertible matrix and a diagonal matrix has distinct eigenvalues.