Let ℋ denote the Hilbert space
of square summable analytic function on the unit disk, and consider those formal
differential operators
which give rise to symmetric operators in ℋ This paper is devoted to a study
of when these operators are actually self-adjoint or admit of self-adjoint
extensions in ℋ. It is shown that in the first order case the operator is always
selfadjoint. For n > 1 sufficient conditions on the pi are obtained for the existence
of self-adjoint extensions. In particular a condition on the coefficients is
obtained which insures that the operator has defect indices equal to the order of
L.
