Vol. 49, No. 2, 1973

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Self-adjoint extensions of symmetric differential operators

Arnold Lewis Villone

Vol. 49 (1973), No. 2, 569–577

Let denote the Hilbert space of square summable analytic function on the unit disk, and consider those formal differential operators

    ∑n    i
L =    piD

which give rise to symmetric operators in This paper is devoted to a study of when these operators are actually self-adjoint or admit of self-adjoint extensions in . 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.

Mathematical Subject Classification 2000
Primary: 47B37
Received: 15 September 1972
Revised: 5 January 1973
Published: 1 December 1973
Arnold Lewis Villone