Operator convex functions of several variables
type: Item
instance of: scholarly article
title: Operator convex functions of several variables
zbMATH Open document ID: 0902.47013
author: Frank-Peter Hansen
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1998-06-16T00:00:00Z
review text: Summary: The functional calculus for functions of several variables associates to each tuple \(x= (x_1,\dots, x_k)\) of selfadjoint operators on Hilbert spaces \(H_1,\dots, H_k\) an operator \(f(x)\) in the tensor product \(B(H_1)\otimes\cdots\otimes B(H_k)\). We introduce the notion of generalized Hessian matrices associated with \(f\). Those matrices are used as the building blocks of a structure theorem for the second Fréchet differential of the map \(x\to f(x)\). As an application, we derive that functions with positive semidefinite generalized Hessian matrices of arbitrary order are operator convex. The result generalizes a theorem of \textit{F. Kraus} [Math. Z. 41, 18-42 (1936; Zbl 0013.39701)] for functions of one variable.
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 15A45
zbMATH DE Number: 1117533
zbMATH Keywords: functional calculus for functions of several variables
zbMATH Keywords: tensor product
zbMATH Keywords: generalized Hessian matrices
zbMATH Keywords: second Fréchet differential
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195145324
OpenAlex ID: W2074082380
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DOI: 10.2977/PRIMS/1195145324
description: scientific article; zbMATH DE number 1117533