A rich structure related to the construction of holomorphic matrix functions

Abstract

The problem of designing controllers that are robust with respect to uncertainty leads to questions that are in the areas of operator theory and several complex variables. One direction is the engineering problem of -synthesis, which has led to the study of certain inhomogeneous domains such as the symmetrised polydisc and the tetrablock. The - synthesis problem involves the construction of holomorphic matrix valued functions on the disc, subject to interpolation conditions and a boundedness condition. In more detail, let 1; : : : ; n be distinct points in the disc, and let W1; : : : ;Wn be 2 2 matrices. The -synthesis problem related to the symmetrised bidisc involves nding a holomorphic 2 2 matrix function F on the disc such that F( j) = Wj for all j, and the spectral radius of F( ) is less than or equal to 1 for all in the disc. The -synthesis problem related to the tetrablock involves nding a holomorphic 2 2 matrix function F on the disc such that F( j) = Wj for all j, and the structured singular value (for the diagonal matrices with entries in C) of F( ) is less than or equal to 1 for all in the disc. For the symmetrised bidisc and for the tetrablock, we study the structure of interconnections between the matricial Schur class, the Schur class of the bidisc, the set of pairs of positive kernels on the bidisc subject to a boundedness condition, and the set of holomorphic functions from the disc into the given inhomogeneous domain. We use the theory of reproducing kernels and Hilbert function spaces in these connections. We give a solvability criterion for the interpolation problem that arises from the -synthesis problem related to the tetrablock. Our strategy for this problem is the following: (i) reduce the -synthesis problem to an interpolation problem in the set of holomorphic functions from the disc into the tetrablock; (ii) induce a duality between this set and the Schur class of the bidisc; and then (iii) use Hilbert space models for this Schur class to obtain necessary and su cient conditions for solvability.