5 research outputs found
A rich structure related to the construction of holomorphic matrix functions
PhD ThesisThe 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
Prime and Semiprime Inner Functions
This paper studies the structure of inner functions under the operation of composition, and in particular the notions or primeness and semiprimeness. Results proved include the density of prime finite Blaschke products in the set of finite Blaschke products, the semiprimeness of finite products of thin Blaschke products and their approximability by prime Blaschke products. An example of a nonsemiprime Blaschke product that is a Frostman Blaschke product is also provided
Prime and semiprime inner functions
Abstract
This paper studies the structure of inner functions under the operation of composition, and in particular the notions or primeness and semiprimeness. Results proved include the density of prime finite Blaschke products in the set of finite Blaschke products, the semiprimeness of finite products of thin Blaschke products and their approximability by prime Blaschke products. An example of a nonsemiprime Blaschke product that is a Frostman Blaschke product is also provided