Non-commutative perspectives. (English) Zbl 1308.47014

Effros, E. G.; Hansen, Frank

Ann. Funct. Anal. 5, No. 2, 74-79 (2014).

A perspective function \(\mathcal P_f\) is a function of two variables of the form \[ \mathcal P_f(t,s)=sf(ts^{-1})\quad (t,s>0)\,, \] where \(f\) is a function defined in \(\mathbb R^{>0}\). If \(A\) and \(B\) are positive definite matrices, then the matrix \[ \mathcal P_f(A,B)=B^{1/2}f\left(B^{-1/2}AB^{-1/2}\right)B^{1/2} \] is well-defined.
In the paper under review, the authors prove that, if \(\mathcal H\) is an infinite dimensional Hilbert space and \(F\) is a regular map from pairs of bounded positive semi-definite operators on \(\mathcal H\) into \(B(\mathcal H)\) such that it is homogeneous, mid-convex, \(F(0,0)=0\) and \(B\mapsto F(1,B)\) is continuous on bounded subset in the strong topology, where \(1\) denotes the unit operator on \(\mathcal H\), then there exists an operator convex function \(f:\mathbb R^{>0}\to \mathbb R\) such that \[ F(1,t\cdot 1)=f(t)1 \quad(t>0)\,, \] and \[ F(A,B)=A^{1/2}f\left(A^{-1/2}BA^{-1/2}\right)A^{1/2} \] for positive definite invertible operators \(A\) and \(B\).

Reviewer: Ali Morassaei (Zanjan)

MSC:

47A56	Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
26B25	Convexity of real functions of several variables, generalizations

Keywords:

perspective; non-commutative; convexity