Abstract:
Let H be an abstract separable Hilbert space. We will consider the Hilbert space H1 whose elements are functions f(x) with domain H and we will also consider the set of self-adjoint operators Q(x) in H of the form Q(x)=A+B(x). In this formula A⩾E, B(x)⩾0, and the operator B(x) is bounded for all x. An operator L0 is defined on the set of finite, infinitely differentiable (in the strong sense) functions y(x)\inH1 according to the formula: L0y=−y″(-\infty<x<\infty). It is proved that the closure of the operator L_0 is a self-adjoint operator in H_1 under the given assumptions.
This publication is cited in the following 4 articles:
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M. M. Gekhtman, “On the spectrum of an operator Sturm–Liouville equation”, Funct. Anal. Appl., 6:2 (1972), 151–152
L. I. Vainerman, M. L. Gorbachuk, “On self-adjoint semibounded abstract differential operators”, Ukr Math J, 22:6 (1971), 694