The Closed Extensions of a Closed Operator

2018/07/10 by Fischbacher, Christoph
#47A05 #47A70 #47B25 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1807.03471

Abstract

Given a densely defined and closed operator A acting on a complex Hilbert space H, we establish a one-to-one correspondence between its closed extensions and subspaces \mathfrakM\subsetD(A^*), that are closed with respect to the graph norm of A^* and satisfy certain conditions. In particular, this will allow us to characterize all densely defined and closed restrictions of A^*. After this, we will express our results using the language of Gel'fand triples generalizing the well-known results for the selfadjoint case. As applications we construct: (i) a sequence of densely defined operators that converge in the generalized sense to a non-densely defined operator, (ii) a non-closable extension of a symmetric operator and (iii) selfadjoint extensions of Laplacians with a generalized boundary condition.

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