On numerical range distortion under metric similarity in Hilbert space operators

Abstract

This paper presents an analysis of numerical range transformations under the equivalence relation of metric similarity in Hilbert spaces. Operators are metrically similar if for some unitary operator and positive invertible operator . We prove that metric similarity constitutes an equivalence relation and establish sharp, optimal bounds for numerical range distortions: with , demonstrating optimality through explicit constructions. The work confirms spectral invariance preservation of and under metric similarity. MSC 2020: Primary 47B15, 47B37; Secondary 47B20, 47A25