## Annals of Functional Analysis

### Conic structure of the non-negative operator convex functions on $(0,\infty)$

#### Abstract

The conic structure of the convex cone of non-negative operator convex functions on $(0,\infty)$ (also on $(-1,1)$) is clarified. We completely determine the extreme rays, the closed faces, and the simplicial closed faces of this convex cone.

#### Article information

Source
Ann. Funct. Anal., Volume 5, Number 2 (2014), 158-175.

Dates
First available in Project Euclid: 7 April 2014

https://projecteuclid.org/euclid.afa/1396833511

Digital Object Identifier
doi:10.15352/afa/1396833511

Mathematical Reviews number (MathSciNet)
MR3192018

Zentralblatt MATH identifier
1305.60103

#### Citation

Franz, Uwe; Hiai, Fumio. Conic structure of the non-negative operator convex functions on $(0,\infty)$. Ann. Funct. Anal. 5 (2014), no. 2, 158--175. doi:10.15352/afa/1396833511. https://projecteuclid.org/euclid.afa/1396833511

#### References

• T. Ando, Topics on Operator Inequalities, Lecture notes (mimeographed), Hokkaido Univ., Sapporo, 1978.
• T. Ando and F. Hiai, Operator log-convex functions and operator means, Math. Ann. 350 (2011), 611–630.
• R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1996.
• W.F. Donoghue, Jr., Monotone Matrix Functions and Analytic Continuation, Springer-Verlag, Berlin-Heidelberg-New York, 1974.
• U. Franz, F. Hiai and E. Ricard, Higher order extension of Löwner's theory: Operator $k$-tone functions, Trans. Amer. Math. Soc. (to appear).
• F. Hansen, Trace functions as Laplace transforms, J. Math. Phys. 47, (2006), 043504, 1–11.
• F. Hansen and G.K. Pedersen, Jensen's inequality for operators and Löwner's theorem, Math. Ann. 258 (1982), 229–241.
• F. Hiai, Matrix Analysis: Matrix Monotone Functions, Matrix Means, and Majorization (GSIS selected lectures), Interdisciplinary Information Sciences 16 (2010), 139–248.
• F. Kraus, Über konvexe Matrixfunktionen, Math. Z. 41 (1936), 18–42.
• K. Löwner, Über monotone Matrixfunktionen, Math. Z. 38 (1934), 177–216.
• R.R. Phelps, Lectures on Choquet theorem, 2nd edition, Lecture Notes in Math., Vol. 1757, Springer-Verlag, 2001.
• B. Simon, Convexity: An Analytic Viewpoint, Cambridge Tracts in Math., Vol. 187, Cambridge University Press, Cambridge, 2011.