By Komarchev I.A.
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Extra resources for 2-Abolutely summable oeprators in certain Banach spaces
Therefore (by Thm. 23 in Ref. 8), it is closable and we denote its closure by qα and its form domain, which is easily seen to be independent of α (see Ref. 4) by H+1 . 1. 3). Then there is a unique self-adjoint extension of H such that the domain of the operator is contained in H+1 × H0 . 1. Note that what this theorem says that “in some sense” the Schur complement of −S is positive, and therefore has a natural self-adjoint extension, then one can define a distinguished self-adjoint extension of the operator H which is unique among those whose domain is contained in the form domain of the Schur complement of −S times H0 .
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4. We believe that this will clarify the precise structure and hypotheses necessary to define distinguished self-adjoint extensions by this method. The main idea in our method is that Hardy-like inequalities are fundamental to define distinguished (physically relevant) self-adjoint extensions even for operators that are not bounded below. We are going to apply our method to operators H defined on D02 , where D0 is some dense subspace of a Hilbert space H0 . 1) H= T −S where all the above operators satisfy Q = T ∗ , P = P ∗ , S = S ∗ and S ≥ c1 I > 0.
2-Abolutely summable oeprators in certain Banach spaces by Komarchev I.A.