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Research Interest

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.