The calculus of operator functions and operator convexity
Volume 390 / 2000
Abstract
The paper is concerned with the Fréchet differentiability and operator convexity of the operator functions on sets of self-adjoint operators on finite-dimensional inner product spaces which are associated with real-valued functions of one or two variables. In Part I it is shown that if a real-valued function is $L$ times continuously differentiable then the associated operator functions are $L$ times Fréchet differentiable with continuous Fréchet derivatives. It is shown that the operator functions corresponding to a real-valued function $f$ can be expressed algebraically in terms of its first Fréchet derivatives. There are then the natural differential conditions for operator monotonicity and convexity and the latter is used in Part II which is concerned with operator convex functions of two variables. The set ${\cal OC}_{2}$ of operator convex functions on $(-1,1)^{2}$ is a convex cone. A three-parameter family of faces $F(\alpha ,\beta ,e)$, of dimension 3 (the trivial case), 4, 6, 7 or 8, of ${\cal OC}_{2}$ is identified and investigated.
