 ## John B Conway

### 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.

### I have collected here a few documents associated with some of my books. These are in pdf file.

1. Corrections for my book "A Course in Functional Analysis" (Second Edition, Third Printing).
2. 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.
3. 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.
4. Additions and Changes for "Functions of One Complex Variable." Some of these are comments on the exercises and some are references to the literature.
5. Corrections for "Functions of One Complex Variable,II."
6. 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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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.  Travel Wine and Food The Wine Advocate The Wine Spectator Time

I recently gave a lecture on clocks and calendars to a class on the Millennium given by Professor Rossalind Hacket of the religious studies department.  You can find the text of my talk together with various links here .

John Conway -- conway@gwu.edu