Let ℋ denote the Hilbert space
of square summable analytic functions on the unit disk, and consider the formal
differential operator
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 EΔ corresponding to the interval
Δ = (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.
