Research Interests
My research interests center on the study of bounded operators on a
Hilbert space, particularly those parts that connect with the theory of
analytic functions. I like to look at problems in operator theory
that are susceptible to an application of complex function theory, and
I have specialized in those operators where this naturally
occurs. A prime example of such operator is the class of
subnormal operators. These are operators that are the restriction
of a normal operator to an invariant subspace. The theory of
normal operators, which is very well understood and essentially
complete, is based on measure theory. Subnormal operators are
asymmetric. One could say that normal operators are to subnormal
operators as continuous functions are to analytic functions.
Typical examples of subnormal operators arise from analytic
functions. One such example is the unilateral shift.
Another is the Bergman shift, defined as follows. Fix a bounded
open set $G$ in the complex plane and let $H$ be the Hilbert space of
all analytic functions on $G$ that are square integrable with respect
to area measure on $G$. Define $S:H\rightarrow H$ by
$(Sf)(z)=zf(z)$ for all $f$ in $H$.
I also have an interest in non-abelian approximation of operators on
Hilbert space. Abelian approximation theory deals with
approximating functions. The underlying idea is that the ring of
bounded operators on a Hilbert space constitutes a non-abelian version
of the ring of continuous, scalar-valued functions on a compact metric
space. A typical problem is, "What is the closure of the set of
operators having a square root?" If the Hilbert space is finite
dimensional, it is possible to characterize which square matrices have
a square root. (A nice application of Jordan forms.) If the
Hilbert space is infinite dimensional, however, such a characterization
is very far from existing. However, you can charaterize which
operators are the limits of operators having a square root, and the
answer is realtively simple to state and aestheically pleasing.
See J B Conway and B B Morrel, ``Roots and logarithms of bounded
operators on a Hilbert space,'' {\sl J Funct Anal} {\bf 70}
(1987) 171--193.
Short Biography
I was born, raised, and educated in New Orleans, La, receiving my BS
from Loyola University in 1961. In 1965 I got a PhD from LSU and
began my career as a mathematician at Indiana University where I
remained until 1990 when I accepted the job as head of the mathematic
at the University of Tennessee. After a brief stay at NSF I arrived at
GWU in 2006. I spent my first sabbatical in 1972 at the Free University
of Amsterdam, and I have spent summers at Berkeley (1968) and the
University of Grenoble (1981).
I have had 19
PhD students.
Here
are some photos of me and my students taken at the SEAM 2000 at the
University of Virginia.
I have collected here a few documents associated with some of my
books. These are in pdf file.
- Corrections for my book "A
Course in Functional Analysis" (Second Edition, Third Printing).
- Notes for a Third Edition of "A
Course in Functional Analysis." . I am not currently planning a
third edition. These notes are changes that are not corrections and are
too extensive to incorporate in another printing.
- Corrections for my book "Functions
of One Complex Variable." (Second edition, fourth printing). The
seventh printing, which exists, incorporates these corrections. I have
not compiled a set of corrections for later printings.
- Additions and Changes for
"Functions of One Complex Variable." Some of these are comments
on the exercises and some are references to the literature.
- Corrections for "Functions
of One Complex Variable,II."
- Corrections for "Theory of
Subnormal Operators."
You are also welcomed to down load the file References
. This is a pdf file of almost all the refernces I have used in
books and papers that I have written.
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A pdf version of my CV is available.
House in France
The renovation of our house in Brittany is complete, including building
the terrace. You are welcomed to browse some pictures of the house.
Family
photos are found here.
Here is a link to the Anglo-American
School in St Petersburg , where my son teaches.
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Below are a few of my non-mathematical interests and some links to
pages I like.