Left-definite theory with applications to orthogonal polynomials

In the past several years, there has been considerable progress made on a general left-definite theory associated with a self-adjoint operator AA that is bounded below in a Hilbert space HH; the term ‘left-definite’ has its origins in differential equations but Littlejohn and Wellman [L. L. Littlejohn, R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181 (2) (2002) 280–339] generalized the main ideas to a general abstract setting. In particular, it is known that such an operator AA generates a continuum {Hr}r>0{Hr}r>0 of Hilbert spaces and a continuum of {Ar}r>0{Ar}r>0 of self-adjoint operators. In this paper, we review the main theoretical results in [L. L. Littlejohn, R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181 (2) (2002) 280–339]; moreover, we apply these results to several specific examples, including the classical orthogonal polynomials of Laguerre, Hermite, and Jacobi.