Convex Entire Noncommutative Functions are Polynomials of Degree Two or Less
| Autor: | J. William Helton, Victor Vinnikov, James Eldred Pascoe, Ryan Tully-Doyle |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
0209 industrial biotechnology
Polynomial Algebra and Number Theory Degree (graph theory) 010102 general mathematics Open set Regular polygon 02 engineering and technology Function (mathematics) 01 natural sciences Noncommutative geometry Combinatorics Matrix (mathematics) 020901 industrial engineering & automation 0101 mathematics Tuple Analysis Mathematics |
| Zdroj: | Integral Equations and Operator Theory. 86:151-163 |
| ISSN: | 1420-8989 0378-620X |
| DOI: | 10.1007/s00020-016-2317-y |
| Popis: | This paper concerns matrix “convex” functions of (free) noncommuting variables, $${x = (x_1, \ldots, x_g)}$$ . 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, $${a = (a_1, \ldots, a_{\tilde{g}})}$$ and $${x = (x_1, \ldots, x_g)}$$ that is both“analytic” and matrix convex in $${x}$$ on a “noncommutative open set” in $${a}$$ is a polynomial of degree two or less. |
| Databáze: | OpenAIRE |
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