- 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 L2(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 L2(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 {Hr }r>0 (Hr is called the rth left-definite space associated with (H,A)) and a continuum of self-adjoint operators {Ar}r>0 (Ar is called the rth 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.