 
Summary: November 16, 2006
POSITIVE POLYNOMIALS IN SCALAR AND MATRIX
VARIABLES, THE SPECTRAL THEOREM AND
OPTIMIZATION
J. WILLIAM HELTON AND MIHAI PUTINAR
Tibi Constantinescu, in memoriam
Abstract. We follow a stream of the history of positive matrices and
positive functionals, as applied to algebraic sums of squares decomposi
tions, with emphasis on the interaction between classical moment prob
lems, function theory of one or several complex variables and modern
operator theory. The second part of the survey focuses on recently dis
covered connections between real algebraic geometry and optimization
as well as polynomials in matrix variables and some control theory prob
lems. These new applications have prompted a series of recent studies
devoted to the structure of positivity and convexity in a free algebra,
the appropriate setting for analyzing inequalities on polynomials having
matrix variables. We sketch some of these developments, add to them
and comment on the rapidly growing literature.
1. Introduction
This is an essay, addressed to nonexperts, on the structure of positive
