Abstract.This paper is devoted to the definition and the study of the generalized inverses of unbounded densely defined closed operators in Banach and Hubert spaces. In this latter case an identity is established that links the orthogonal projection on the graph of an operator to the orthogonal projection on the graph of its Moore-Penrose inverse.
The selfadjoint extensions of a closed linear relation R from a Hilbert space H 1 to a Hilbert space H 2 are considered in the Hilbert space H 1 ⊕ H 2 that contains the graph of R. They will be described by 2 × 2 blocks of linear relations and by means of boundary triplets associated with a closed symmetric relation S in H 1 ⊕ H 2 that is induced by R. Such a relation is characterized by the orthogonality property dom S ⊥ ran S and it is nonnegative. All nonnegative selfadjoint extensions A, in particular the Friedrichs and Kreȋn-von Neumann extensions, are parametrized via an explicit block formula. In particular, it is shown that A belongs to the class of extremal extensions of S if and only if dom A ⊥ ran A. In addition, using asymptotic properties of an associated Weyl function, it is shown that there is a natural correspondence between semibounded selfadjoint extensions of S and semibounded parameters describing them if and only if the operator part of R is bounded.
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