## » Research

- Spectral Theory of Differential Operators, General Functional Analysis and Operator Theory
- Orthogonal Polynomials and Special Functions
- Inequalities

Currently, I work in abstract and applied operator theory as well as the applications of these areas to special functions, particularly orthogonal polynomials.

Here's a brief description of what I am currently working on in my research. The Glazman-Krein-Naimark (GKN) theory provides a *recipe*, so to speak, for determining all self-adjoint extensions in the Hilbert space L^{2}(I;*w) *of a given formally Lagrangian symmetrizable differential expression with symmetry factor *w*. For example, it is this theory that is used to determine what the appropriate boundary conditions are that define the self-adjoint differential operators generated by the classical second-order differential expressions of Jacobi, Laguerre, and Hermite which have the corresponding orthogonal polynomials as eigenfunctions. Together with several colleagues, I am working on extending this GKN theory beyond L^{2}(I;*w)* and into Sobolev spaces. It turns out that, if we somewhat *relax* the parameters alpha and/or beta in the Jacobi or Laguerre differential expressions, the corresponding polynomial solutions to these differential equations will be Sobolev orthogonal. It is natural to ask: what is the corresponding self-adjoint operator in the corresponding Sobolev space? We have worked out most of these new Sobolev examples - now we want to construct a general theory around them.

I am also working on further extending, and applying, a general left-definite theory that Richard Wellman and I have developed. Left-definite theory (the notation is due to Schäfke and Schneider) has its roots in earlier work of Hermann Weyl, specifically on boundary value problems involving second-order differential equations. We have taken an abstract approach and have shown that, given any self-adjoint operator A in a Hilbert space H that is bounded below by a positive constant, there exists a continuum of Hilbert spaces {H_{r
} }_{r>0} (H_{r} is called the r^{th} left-definite space associated with (H,A)) and a continuum of self-adjoint operators {A_{r} }_{r>0} (A_{r} is called the r^{th} left-definite operator associated with (H,A)). Quite surprisingly, these left-definite spaces and operators reveal new information about the original operator A and its powers. Together with my Ph.D. students, we are applying this theory to self-adjoint difference and differential operators.