The research of this proposal is in functional analysis and operator theory related to engineering system theory and to several variables and mutliply connected domains in the complex plane. Many problems in linear control theory can be formulated in terms of polynomials in several non-commuting variables. In applications, convexity of the polynomial is either known or desired. A sample result of Helton and the PI is that a convex polynomial in non-commuting variables has degree at most two and can be written as a linear term plus a sum of squares of linear terms. McCullough will continue extending the theory of non-commutative polynomials and rational functions with an eye toward convexity. He will also continue to investigate non-self-adjoint operator algebras of functions, particularly those associated to multiply connected domains and to several commuting variables whose unit balls are specified by a given collection of functions.
Ideas and techniques from functional analysis and operator theory are powerful tools in the study of matrix inequalities which take the form of a polynomial or rational function of matrices (as the variables) being positive semi-definite. Such matrix inequalities model a large class of engineering linear systems problems, like those arising in the design and control of automatic controllers. Convex matrix inequalities are important for design and numerical implementation. A goal of this proposal is to further understand convex matrix inequalities and when and how it is possible to convert a non-convex inequality into a convex inequality.
