Recently Scott McCullough and I solved the rational dilation problem
for two holed domains. A subset *X* of the complex plane is
called a spectral set for an operator *T* if von Neumann's
inequality holds for all rational functions in *T* with poles
off of *X*. The operator *T* is said to have a normal
boundary of *X* dilation if such rational functions in
*T* can be written as the compression of the same rational
function in a normal operator with spectrum on the boundary of
*X*. It is not difficult to see that if *T* has a
normal boundary of *X* dilation, then *X* is a spectral
set for *T*. On the other hand, the Sz.-Nagy dilation theorem
implies that the converse is true when *X* is the unit disk. A
much more difficult result of Agler's from 1985 showed that again the
converse is true if *X* is an annulus. This then implies the
result for any doubly connected region with distinct boundary
components which are Jordan arcs. Recently, Agler, Harland and
Raphael announced an example of a two holed domain and a 4x4 matrix on
that domain for which rational dilation fails, based on numerical
calculations. Scott McCullough and I have shown that this is in fact
a universal property of two holed domains with boundary components
which are disjoint Jordan arcs. A copy of the preprint is available
here.

Stefania Marcantognini spent a year in 2004-2005 as a visitor in the School. With Scott McCullough we wrote a paper on interpolation on semigroupoid algebras (available here). This provides a framework encompassing a wide variety of interpolation results. The salient feature of such algebras is that the product generalises both the pointwise and convolution products, and was in a more restrictive sense first considered for interpolation problems by Michael Jury. Here we also consider families of test functions which are in duality with a collection of reproducing kernels. These kernels give a norm on the functions over the semigroupoid, and with this norm the test functions delineate the unit ball. In many cases of interest a single reproducing kernel is insufficient to describe the norm on the function algebra (this is the case, for example, with H-infinity of the bidisk or of an annulus). So while the multiplier algebras of reproducing kernel Hilbert spaces are interesting, they are too restrictive to cover such important examples. A particularly satisfying aspect of the work is the realization formula for elements of the unit ball of a semigroupoid algebra, which is analogous to that for H-infinity of the disk. We also consider interpolation problems and show that a condition along the lines of that found classically in Nevanlinna-Pick interpolation is valid.

Somewhat older recent work has mostly been on abstract model theory for families of operators, and families of representations of algebras, as formulated by Jim Agler. Applications have included the study of model theory for the families of hyponormal contractions and -contractions. This work has been done in collaboration with Scott McCullough and Hugo Woerdeman. Scott McCullough and I have also looked at the problem raised by Arveson in his seminal papers from the late 60's and early 70's on the existence of boundary representations for operator algebras--a sort of noncommutative analogue of the Shilov boundary for function algebras. We were able to show that if the condition of irreducibility is dropped, then such representations always exist, giving a direct proof of Hamana's result on the existence of C*-envelopes for unital operator algebras.

I am also interested in Positivstellensätze (problems involving expressing positive things as sums of squares). In particular, I have worked on the operator Fejer-Riesz lemma on factorisation of nonnegative trigonometric polynomials as sums of "squares" of analytic polynomials, both in one and several variables. This has applications in filter design, H-infinity control, and wavelet theory among other things. In a recent paper with Hugo Woerdeman, we consider the question of what it means for a function in several variables to be outer, since equivalent definitions in the single variable case correspond to different things when more than one variable is involved.

Other areas of research interest include operator theory on Krein spaces (a type of indefinite metric space). This has primarily centered around commutant lifting and matrix completion problems, though with brief forays into invariant subspaces (for example, using Krein space techniques to get new invariant subspace results for products of selfadjoint operators or compact perturbations of selfadjoint operators on Hilbert space). Parts of this work was in collaboration with James Rovnyak.

Michael A. Dritschel Department of Mathematics School of Mathematics and Statistics University of Newcastle-upon-Tyne Newcastle upon Tyne NE1 7RU England |
Home page:
http://www.mas.ncl.ac.uk/~nmad1/
School of Mathematics and Statistics web page: http://www.ncl.ac.uk/math/ Newcastle University's web page: http://www.ncl.ac.uk/ Email: M.A.Dritschel ©2003, Michael A. Dritschel |

Last modified: Thu Apr 16 12:05:15 BST 2009