ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 47B15 (26B25) Brown, A. L. , Vasudeva, H. L., The Pnajab University, India Operator convex functions of two variables To an f : (−1, 1)^{2} → ℝ there correspond operator functions f : S_{(−1,1)}(ℂ^{m}) × S_{(−1,1)}(ℂ^{N}) → S(ℂ^{M} ⊗ ℂ^{N}) where S(X) denotes the set of self-adjoint operators on the inner product space X, S_{(−1,1)}(X) those self-adjoint operators with spectrum in (- 1,1), and M, N ∈ ℕ. Then f is said to be operator convex if, for each (M, N) ∈ ℕ^{2}, f((1 − θ)V + θW) ≤ (1−θ)f(V)+θf(W) for all V, W ∈ S_{(−1,1)}(ℂ^{M})×S_{(−1,1)}(ℂ^{n}) and θin[−1, 1]. The set OC_{2} of operator convex functions on (−1, 1)^{2} is a convex cone. A three parameter family of non-trivial finite dimensional faces of OC_{2} is identified. This contrasts with the one variable situation (v. F. Hansen and G. K. Pedersen, Jensen’s inequality for operators and L“owner’s Theorem, Math. Ann. 258: 229-241, 1982) in which the set of extreme points of a base of the convex cone of operator convex functions g : (−1, 1) → ℝ form a topological interval.