Self-adjoint differential operators

Abstract

Let H denote the Hilbert space of square summable analytic functions on the unit disk, and consider the formal differential operator (Formula Presented) where the pi are in H. This paper is devoted to a study of symmetric operators in H arising from L. A characterization of those L which give rise to symmetric operators S is obtained, and the question of when such an S is selfadjoint or admits of a self-adjoint extension is considered. If A is a self adjoint extension of S and E(λ) the associated resolution of the identity, the projection E∆ corresponding to the interval ∆ = (a, b] is shown to be an integral operator whose kernel can be expressed in terms of a basis of solutions for the equation (L — l)u = 0 and a spectral matrix. © 1970, Pacific Journal of Mathematics.