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2011

The fast track to L\"owner's theorem

Hansen, Frank

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The operator monotone functions defined in the positive half-line are of particular importance.

We give a version of the theory in which integral representations for these functions can be established directly without invoking L\"owner's detailed analysis of matrix monotone functions of a fixed order or the theory of analytic functions.
We found a canonical relationship between positive and arbitrary operator monotone functions defined in the positive half-line, and this result effectively reduces the theory to the case of positive functions.
MSC2010 classification: 26A48; 26A51; 47A63.

The fast track to L\"owner's theorem · Around

Built on

Über monotone Matrixfunktionen
K. Löwner · 1934
Earlier work this paper cites.
Über konvekse Matrixfunktionen
F. Kraus · 1936
Earlier work this paper cites.
Significance of Loewner’s theorem in the quantum theory of collisions
E.P. Wigner and J.v. Neumann · 1954
Earlier work this paper cites.
Monotone and convex operator functions
J. Bendat and S. Sherman · 1955
Earlier work this paper cites.
On a theorem of Löwner and its connection with resolvents of self-adjoint transformations
A. Korányi · 1956
Earlier work this paper cites.
Monotone matrix functions and analytic continuation
W. Donoghue · 1974
Earlier work this paper cites.

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