TY - JOUR

T1 - Convex Entire Noncommutative Functions are Polynomials of Degree Two or Less

AU - Helton, J. William

AU - Pascoe, J. E.

AU - Tully-Doyle, Ryan

AU - Vinnikov, Victor

N1 - Funding Information:
J. W. Helton: Partially supported by NSF grant DMS-1201498 and the Ford Motor Co. J. E. Pascoe: Partially supported by NSF grant DMS-1361720. V. Vinnikov: Partially supported by the Israel Science Foundation (Grant No. 322/00).
Publisher Copyright:
© 2016, Springer International Publishing.

PY - 2016/10/1

Y1 - 2016/10/1

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=84990866012&partnerID=8YFLogxK

U2 - 10.1007/s00020-016-2317-y

DO - 10.1007/s00020-016-2317-y

M3 - Article

AN - SCOPUS:84990866012

SN - 0378-620X

VL - 86

SP - 151

EP - 163

JO - Integral Equations and Operator Theory

JF - Integral Equations and Operator Theory

IS - 2

ER -