Abstract
New properties of operator connections and means are established. Specifically, representations of an arbitrary connection by means of a concave representing function, an estimate of the norm of a connection in a von Neumann-Schatten ideal, a relation between operator means and convolutions onto operator domains are obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
W. N. Anderson and B. J. Duffin, “Series and parallel additions of matrices,” J. Math. Anal. Appl.,26, 576–594 (1969).
P. A. Filmore and J. P. Williams, “On operator ranges,” Adv. Math.,7, No. 3, 254–281 (1971).
W. N. Anderson and G. E. Trapp, “Shorted operators,” SIAM J. Appl. Math.,28, No. 1, 60–71 (1975).
É. L. Pekarev and Yu. L. Shmul'yan, “Parallel addition and parallel subtraction of operators,” Izv. Akad. Nauk SSSR, Ser. Mat.,40, No. 2, 366–387 (1976).
T. Ando, “Topics on operator inequalities,” Ryukyu Univ. Lect. Note Ser., No. 1, 44–70 (1978).
W. Pusz and S. L. Woronowics, “Functional calculus for sesquilinear forms and the purification map,” Rep. Math. Phys.,8, No. 2, 159–170 (1975).
W. N. Anderson, T. D. Morley, and G. E. Trapp, “Characterization of parallel subtraction,” Proc. Nat. Acad. Sci. USA,76, 3599–3601 (1979).
F. Kubo and T. Ando, “Means of positive linear operators,” Mathematische Annalen,246, No. 3, 205–224 (1980).
J. Bendat and S. Sherman, “Monotone and convex operator functions,” Trans. Am. Math. Soc.,79, No. 1, 58–71 (1955).
J. I. Fujii, “Operator-concave functions and means of positive linear functionals,” Math. Jpn.,25, No. 4, 453–461 (1980).
T. Ando, “Fixed points of certain maps on positive semidefinite operators,” in: Functional Analysis and Approximation, Proc. Conf. at Oberwolfach, Aug. 9–16, 1980, Basel (1981), pp. 29–38.
T. D. Morley, “Parallel summation, Maxwell's principle and the infimum of projections,” J. Math. Anal. Appl.,70, No. 1, 33–41 (1979).
M. G. Krein, “The theory of self-adjoint extensions of semibounded Hermitian operators, and its applications,” I, Mat. Sb.,20, No. 3, 431–490 (1947).
Yu. L. Shmul'yan, “Hellinger's operator integral,” Mat. Sb.,49, No. 4, 381–430 (1959).
K. Nishio and T. Ando, “Characterizations of operations derived from network connections,” J. Math. Anal. Appl.,53, 539–549 (1976).
T. Ando, “Lebesgue-type decomposition of positive operators,” Acta Sci. Math., Szeged, 38, 253–260 (1976).
É. L. Pekarev, “On the convolution onto an operator domain,” Funkts. Anal. Prilozhen.,12, No. 3, 84–85 (1978).
J. I. Fujii, “Initial conditions on operator-monotone functions,” Math. Jpn.,24, No. 4, 459–462 (1979).
Author information
Authors and Affiliations
Additional information
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 723–730, June, 1990.
Rights and permissions
About this article
Cite this article
Arlinskii, Y.M. Theory of operator means. Ukr Math J 42, 639–645 (1990). https://doi.org/10.1007/BF01058906
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01058906