The calculus of operator functions and operator convexity

A. L. Brown, and H. L. Vasudeva. The calculus of operator functions and operator convexity. 2000. <http://eudml.org/doc/285975>.

@book{A2000,
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 𝓞 𝓒₂ of operator convex functions on (-1,1)² is a convex cone. A three-parameter family of faces F(α,β,e), of dimension 3 (the trivial case), 4, 6, 7 or 8, of 𝓞 𝓒₂ is identified and investigated.},
author = {A. L. Brown, H. L. Vasudeva},
keywords = {self-adjoint operator; convex function; convex cone; face; operator monotone function; Fréchet differentiability; operator convex functions},
language = {eng},
title = {The calculus of operator functions and operator convexity},
url = {http://eudml.org/doc/285975},
year = {2000},
}

TY - BOOK
AU - A. L. Brown
AU - H. L. Vasudeva
TI - The calculus of operator functions and operator convexity
PY - 2000
AB - 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 𝓞 𝓒₂ of operator convex functions on (-1,1)² is a convex cone. A three-parameter family of faces F(α,β,e), of dimension 3 (the trivial case), 4, 6, 7 or 8, of 𝓞 𝓒₂ is identified and investigated.
LA - eng
KW - self-adjoint operator; convex function; convex cone; face; operator monotone function; Fréchet differentiability; operator convex functions
UR - http://eudml.org/doc/285975
ER -