Abstract
In this note, we characterize all those bijective transformations on the set of positive semidefinite matrices which preserve an arbitrary unitarily invariant norm of any given Kubo–Ando mean corresponding to an n-monotone strictly concave generating function. This result strengthens a former theorem of Molnár and Szokol.
Similar content being viewed by others
References
R. Bhatia, Matrix Analysis, Springer-Verlag (New York, 1997).
P. Busch and S. P. Gudder, Effects as functions on projective Hilbert space, Lett. Math. Phys., 47 (1999), 329–337.
F. Hansen, G. Ji and J. Tomiyama, Gaps between classes of matrix monotone functions, Bull. Lond. Math. Soc., 36 (2004), 53–58.
F. Hiai and D. Petz, Riemannian metrics on positive definite matrices related to means, Linear Algebra Appl., 430 (2009), 3105–3130.
F. Hiai and D. Petz, Introduction to Matrix Analysis and Applications, Springer (Switzerland, 2014).
D. T. Hoa, T. M. Ho and H. Osaka, Interpolation classes and matrix means, Banach J. Math. Anal., 9 (2015), 140–152.
F. Kubo and T. Ando, Means of positive linear operators, Math. Ann., 246 (1980), 205–224.
R. Mathias, Concavity of monotone matrix functions of finite order, Linear Multilinear Algebra, 27 (1990), 129–138.
L. Molnár, Order-automorphisms of the set of bounded observables, J. Math. Phys., 42 (2001), 5904–5909.
L. Molnár and P. Szokol, Transformations preserving norms of means of positive operators and nonnegative functions, Integral Equations Operator Theory, 83 (2015), 271–290.
C. Niculescu and L.-E. Persson, Convex Functions and Their Applications. A Contemporary Approach, CMC Books in Mathematics, Springer (New York, 2006).
H. Osaka, S. Silvestrov and J. Tomiyama, Monotone operator functions, gaps and power moment problem, Math. Scand., 100 (2007), 161–183.
D. Petz, Monotone metrics on matrix spaces, Linear Algebra Appl., 244 (1996), 81–96.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Research, Development and Innovation Office – NKFIH Reg. No. K115383.
Rights and permissions
About this article
Cite this article
Gaál, M., Nagy, G. A Characterization of Unitary-Antiunitary Similarity Transformations via Kubo–Ando Means. Anal Math 45, 311–319 (2019). https://doi.org/10.1007/s10476-018-0401-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10476-018-0401-z