The numerical range of a contraction with finite defect numbers

TL;DR

This paper characterizes the numerical range of a contraction with finite defect numbers, showing it as an intersection of dilations' numerical ranges and exploring its geometric properties.

Contribution

It provides a new description of the numerical range for contractions with finite defect numbers and details geometric properties when the defect number is one.

Findings

Numerical range equals intersection of dilations' ranges for finite defect contractions.

Detailed geometric properties of the numerical range when defect number is one.

Characterization of the numerical range closure in terms of dilations.

Abstract

An n-dilation of a contraction T acting on a Hilbert space H is a unitary dilation acting on H \oplus C^n. We show that if both defect numbers of T are equal to n, then the closure of the numerical range of T is the intersection of the closures of the numerical ranges of its n-dilations. We also obtain detailed information about the geometrical properties of the numerical range of T in case n=1.