The spectral Nevanlinna-Pick problem
by
N. J. Young
Newcastle University
Coauthors: Jim Agler (University of California at San Diego)

The H^\infty approach to the robust stabilisation of uncertain systems leads to problems which are generalisations of well known problems in function theory. One problem of this type is called \mu-synthesis. It is a broadly formulated interpolation problem. A much-studied special case is the spectral Nevanlinna-Pick problem:

Given points z₁, ... , z_n in the open unit disc D and k ×k matrices W₁, ..., W_n, determine whether there exists an analytic function F : --> M_n such that F(z_i) = W_i for 1 <= i <= n and r(F(z)) <= 1 for all z in D.

Here r(.) denotes the spectral radius. I will discuss this problem, particularly for the case of 2 ×2 matrices. Operator-theoretic methods lead to a necessary condition of Pick type for the existence of F; in the case of two interpolation points, this condition is also sufficient. On the way we obtain a novel Schwarz Lemma for the symmetrised bidisc

\Gamma = def   { (z1+z2, z1 z2) : |z1| <= 1, |z2| <= 1 }.

Date received: May 29, 2000