Convex Entire Noncommutative Functions are Polynomials of Degree Two or Less

J. William Helton; J. E. Pascoe; Ryan Tully-Doyle; Victor Vinnikov

doi:10.1007/s00020-016-2317-y

Convex Entire Noncommutative Functions are Polynomials of Degree Two or Less

J. William Helton, J. E. Pascoe, Ryan Tully-Doyle, Victor Vinnikov

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

This paper concerns matrix “convex” functions of (free) noncommuting variables, x= (x₁, … , x_g). It was shown in Helton and McCullough (SIAM J Matrix Anal Appl 25(4):1124–1139, 2004) that a polynomial in x which is matrix convex is of degree two or less. We prove a more general result: that a function of x that is matrix convex near 0 and also that is “analytic” in some neighborhood of the set of all self-adjoint matrix tuples is in fact a polynomial of degree two or less. More generally, we prove that a function F in two classes of noncommuting variables, a= (a₁, … , a_g~) and x= (x₁, … , x_g) that is both“analytic” and matrix convex in x on a “noncommutative open set” in a is a polynomial of degree two or less.

Original language	English
Pages (from-to)	151-163
Number of pages	13
Journal	Integral Equations and Operator Theory
Volume	86
Issue number	2
DOIs	https://doi.org/10.1007/s00020-016-2317-y
State	Published - 1 Oct 2016

ASJC Scopus subject areas

Analysis
Algebra and Number Theory

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abstract = "This paper concerns matrix “convex” functions of (free) noncommuting variables, x= (x1, … , xg). It was shown in Helton and McCullough (SIAM J Matrix Anal Appl 25(4):1124–1139, 2004) that a polynomial in x which is matrix convex is of degree two or less. We prove a more general result: that a function of x that is matrix convex near 0 and also that is “analytic” in some neighborhood of the set of all self-adjoint matrix tuples is in fact a polynomial of degree two or less. More generally, we prove that a function F in two classes of noncommuting variables, a= (a1, … , ag~) and x= (x1, … , xg) that is both“analytic” and matrix convex in x on a “noncommutative open set” in a is a polynomial of degree two or less.",

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AB - This paper concerns matrix “convex” functions of (free) noncommuting variables, x= (x1, … , xg). It was shown in Helton and McCullough (SIAM J Matrix Anal Appl 25(4):1124–1139, 2004) that a polynomial in x which is matrix convex is of degree two or less. We prove a more general result: that a function of x that is matrix convex near 0 and also that is “analytic” in some neighborhood of the set of all self-adjoint matrix tuples is in fact a polynomial of degree two or less. More generally, we prove that a function F in two classes of noncommuting variables, a= (a1, … , ag~) and x= (x1, … , xg) that is both“analytic” and matrix convex in x on a “noncommutative open set” in a is a polynomial of degree two or less.

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Convex Entire Noncommutative Functions are Polynomials of Degree Two or Less

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