Abstract
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.
Similar content being viewed by others
References
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)
Botelho, F., Jamison, J., Molnár, L.: Surjective isometries on Grassmann spaces. J. Funct. Anal. 265, 2226–2238 (2013)
Busch, P., Gudder, S.P.: Effects as functions on projective Hilbert spaces. Lett. Math. Phys. 47, 329–337 (1999)
Cassinelli, G., De Vito, E., Lahti, P., Levrero, A.: A theorem of Ludwig revisited. Found. Phys. 30, 1757–1763 (2000)
Das, M., Sahoo, M.: Positive Toeplitz operators on the Bergman space. Ann. Funct. Anal. 4, 171–182 (2013)
Dolinar, G., Molnár, L.: Isometries of the space of distribution functions with respect to the Kolmogorov-Smirnov metric. J. Math. Anal. Appl. 348, 494–498 (2008)
Douglas, R.G.: On majorization, factorization, and range inclusion of operators on Hilbert space. Proc. Am. Math. Soc. 17, 413–415 (1966)
Fillmore, P.A., Williams, J.P.: On operator ranges. Adv. Math. 7, 254–281 (1971)
Kadison, R.V.: Order properties of bounded self-adjoint operators. Proc. Am. Math. Soc. 2, 505–510 (1951)
Kadison, R.V.: A generalized Schwarz inequality and algebraic invariants for operator algebras. Ann. Math. 56, 494–503 (1952)
Mankiewicz, P.: On extension of isometries in normed linear spaces. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 20, 367–371 (1972)
Molnár, L.: Order-automorphisms of the set of bounded observables. J. Math. Phys. 42, 5904–5909 (2001)
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)
Molnár, L.: Maps preserving the geometric mean of positive operators. Proc. Am. Math. Soc. 137, 1763–1770 (2009)
Molnár, L.: Maps preserving the harmonic mean or the parallel sum of positive operators. Linear Algebra Appl. 430, 3058–3065 (2009)
Molnár, L.: Maps preserving general means of positive operators. Electron. J. Linear Algebra 22, 864–874 (2011)
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)
Molnár, L., Szokol, P.: Transformations preserving norms of means of positive operators and nonnegative functions. Integral Equ. Oper. Theory 83, 271–290 (2015)
Šemrl, P.: Symmetries of Hilbert space effect algebras. J. Lond. Math. Soc. 88, 417–436 (2013)
Acknowledgements
The author is grateful to the referee for his/her comments which have helped to improve the presentation of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research was supported by the “Lendület” Program (LP2012-46/2012) of the Hungarian Academy of Sciences and by the National Research, Development and Innovation Office NKFIH, Grant No. K115383.
Rights and permissions
About this article
Cite this article
Molnár, L. Busch–Gudder metric on the Cone of Positive Semidefinite Operators and Its Isometries. Integr. Equ. Oper. Theory 90, 20 (2018). https://doi.org/10.1007/s00020-018-2443-9
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s00020-018-2443-9