Vol. 35, No. 2, 1970

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Self-adjoint differential operators

Arnold Lewis Villone

Vol. 35 (1970), No. 2, 517–531

Let denote the Hilbert space of square summable analytic functions on the unit disk, and consider the formal differential operator

    ∑n    i
L =    piD

where the pi are in . This paper is devoted to a study of symmetric operators in 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.

Mathematical Subject Classification
Primary: 47.60
Received: 20 February 1970
Published: 1 November 1970
Arnold Lewis Villone