Some Positive Definite Differential Operators
William Norrie Everitt · Journal of the London Mathematical Society · 1968
1. This paper continues the method developed by the author in [3] in which certain results about the second order differential expression L were obtained where Lij/{x) =- (p(x)fj^+q(x) i^x) [0 ^ x 0, (ii) p is absolutely continuous on [0, X] for all X> 0, (iii) p(x)> 0 for all xe [0, oo). It was shown in [3] that if the additional condition (iv) q is bounded below on [0, oo) is satisfied then the differential expression L is in the limit-point case at infinity, i.e., the differential equation L\\] / = Ai ̂ § has only one linearly independent solution, for each X such thatim X ¥ = 0, which belongs to the class S£2 [0, oo) (see, for example [6; Chapter II §2.1] or [2; Chapter 9]). It will be shown in the next section that the results in [3] imply that certain differential operators in the Hilbert space <£?2[0, oo), generated by the differential expression L,