Numerical Range Distortion Under Unitary Equivalence

Abstract

This paper investigates the preservation properties of numerical ranges under unitary equivalencen transformations in operator theory. We establish that if operators T and S are unitarily equivalent via S = U ∗ TU , then their numerical ranges are identical: W (S) = W (T ). Beyond this fundamental equality, we prove that unitary equivalence preserves critical geometric properties of numerical ranges, including extreme points, exposed points, supporting lines, contact points, and the geometric multiplicity of boundary points. For normal operators, we demonstrate that numerical range equality characterizes unitary equivalence, providing a geometric criterion for this algebraic relation. Additionally, we show that the curvature of the boundary of numerical ranges remains invariant under unitary transformations. These results highlight the deep connection between the algebraic structure of operators and the geometric properties of their numerical ranges, contributing to our understanding of operator behavior under unitary equivalence.