Abstract
We investigate the correspondence between operator monotone functions and Stieltjes functions. Then we shall prove that the class of nonnegative operator monotone functions is closed under certain operations, and applying those we decide the case when some binary operations become operator means.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
N. I. Akhiezer, The classical moment problem, Oliver and Boyd, Edinburgh, 1965.
T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl. 26 (1979), 203–241.
T. Ando, Comparison of norms |||f(A) - f(B)||| and |||f(|A - B|)|||, Math. Z. 197 (1988), 403–409.
C. Berg, The Stieltjes cone is logarithmically convex, Complex Analysis, Joensuu 1978, pp. 46–54, Springer, Berlin-Heidelberg-New York, 1979.
C. Berg, J. P. R. Christensen and P. Ressel, Harmonic analysis on semigroups, Springer-Verlag, New York-Berlin-Heidelberg-Tokyo, 1984.
W. F. Donoghue, Monotone matrix functions and analytic continuation, Springer-Verlag, Berlin-Heidelberg-New York, 1974.
W. F. Donoghue, Communication to T. Ando.
M. G. KreÄn and A. A. Nudel’man, The Markov moment problem and extremal problems, American Mathematical Society, Providence, R.I., 1977.
F. Kubo, Equations of operator means, preprint.
F. Kubo and T. Ando, Means of positive linear operators, Math. Ann. 246 (1980), 205–224.
M. E. Mays, Functions which parametrize means, Amer. Math. Monthly 90 (1983), 679–683.
M. Rosenblum and J. Rovnyak, Hardy classes and operator theory, Oxford University Press, New York, 1985.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to Professor I. Gohberg on the occasion of his 60th birthday
Rights and permissions
Copyright information
© 1989 Birkhäuser Verlag Basel
About this chapter
Cite this chapter
Nakamura, Y. (1989). Classes of Operator Monotone Functions and Stieltjes Functions. In: Dym, H., Goldberg, S., Kaashoek, M.A., Lancaster, P. (eds) The Gohberg Anniversary Collection. Operator Theory: Advances and Applications, vol 40/41. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-9144-8_39
Download citation
DOI: https://doi.org/10.1007/978-3-0348-9144-8_39
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9924-6
Online ISBN: 978-3-0348-9144-8
eBook Packages: Springer Book Archive