Busch–Gudder metric on the Cone of Positive Semidefinite Operators and Its Isometries
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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.
KeywordsPositive cone Positive operators Strength functions Busch–Gudder metric Operator norm Isometries
Mathematics Subject ClassificationPrimary 47B49 47B65 Secondary 54G99
The author is grateful to the referee for his/her comments which have helped to improve the presentation of the paper.
- 1.Ando, T.: Problem of infimum in the positive cone. In: Rassias, T.M., Srivastava, H.M. (eds.) Analytic and Geometric Inequalities and Applications. Mathematics and Its Applications, vol. 478, pp. 1–12. Kluwer Academic Publishers, Dordrecht (1999)Google Scholar
- 11.Mankiewicz, P.: On extension of isometries in normed linear spaces. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 20, 367–371 (1972)Google Scholar
- 13.Molnár, L.: Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces. Lecture Notes in Mathematics, vol. 1895, p. 236. Springer (2007)Google Scholar
- 17.Molnár, L.: The arithmetic, geometric and harmonic means in operator algebras and transformations among them. In: Botelho, F., King, R., Rao, T.S.S.R.K. (eds.) Recent Methods and Research Advances in Operator Theory, Contemporary Mathematics, vol. 687, pp. 193–207. American Mathematical Society, Providence (2017)Google Scholar