Oleg Szehr, Cambridge University, UK

Model theoretic methods and Nevanlinna-Pick interpolation in matrix analysis

Abstract

One of the most basic tasks in matrix analysis is to find a spectral estimate to the norm of a function of a matrix. Examples of this task abound in ubiquitous forms, e.g. as bounds on condition numbers in (numerical) stability analysis, as convergence estimates in the theory of Markov chains and as so-called spectral-variation estimates in theoretical matrix analysis. In this talk we present a method that relates the problem of finding an eigenvalue bound to a function of an operator of certain class to a Nevanlinna-Pick interpolation problem in an associated function algebra. This method draws on deep results from Fourier analysis, interpolation theory and the theory of Hilbert function spaces (e.g. Sarasons’s commutant lifting theorem) and allows us to improve on known spectral estimates in the mentioned examples. For the class of operators on Hilbert space whose norm is bounded by 1 (i.e. contractions) this method provides us with a complete solution to the problem of finding a spectral estimate. A crucial role in this solution is played by the so-called model operator, which is the compression of a backward shift operator on the Hardy space H² to an invariant subspace. We contribute to the theory of such operators by providing explicit matrix representations.