By Komarchev I.A.

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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 .

Linear connections on matrix geometries. Classical Quantum Gravity, 12, no. 6, 1429–1440 (1995). , Linear connections in noncommutative geometry. Classical Quantum Gravity, 12, no. 4, 965–974 (1995). , Tensors and metrics on almost commutative algebras. Lett. Math. Phys. (submitted in 2007). , Levi-Civita connection on almost commutative algebras. Int. J. Geom. Methods Mod. , 4, no. 7, 1075–1085 (2007). , Noncommutative geometry and quantization. A. ´ et al. ), Particles and Fields (Aguas de Lindoia, 1999), World Sci.

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.

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