Busch–Gudder metric on the Cone of Positive Semidefinite Operators and Its Isometries

In this paper we introduce a new distance measure called Busch–Gudder metric on the cone of all positive semidefinite operators acting on a complex Hilbert space. It is defined as the sup-distance between the so-called strength functions corresponding to positive semidefinite operators. We investigate the properties of that metric, among others its relation to the metric induced by the operator norm. We show that in spite of many dissimilarities between the topological features of those two metrics, their isometry groups still coincide.