A class of symmetric differential operators with deficiency indices (1, 1)

Arnold Lewis Villone

Vol. 59 (1975), No. 1, 295–301

DOI: 10.2140/pjm.1975.59.295

Abstract

Let ℋ denote the Hilbert space of analytic functions on the unit disk which are square summable with respect to the usual area measure. In this paper we show that every symmetric differential operator of order two or more having the form L = ∑ _t=0ⁿ(a_i+1(i)zⁱ⁺¹ + a_i−1(i)zⁱ⁻¹)D^t,a₋₁(0) = 0, has defect indices (1, 1) and hence has self-adjont extensions in ℋ. We are also able to show that L + M has defect indices (1,1) where M is a symmetric Euler operator of order n− 1,M = sum_i=0ⁿ⁻¹b_izⁱDⁱ, provided that |b_n−1| < (n − 1)|a_n+1(n)|.

Mathematical Subject Classification 2000

Primary: 47E05

Secondary: 34B25

Milestones

Received: 1 April 1975

Published: 1 July 1975