Geometry of Rigged Hilbert Spaces

Abstract

In this chapter we study extensions of symmetric non-densely defined operators in the triplets H+⊂ H ⊂H- of rigged Hilbert spaces. The Krasnoselskiĭi formulas discussed in Section 1.7 are based upon the indirect decomposition (1.33), where deficiency subspaces and the domain of symmetric operator may be linearly dependent. Introduction of the rigged Hilbert spaces allows us to obtain the direct decomposition and parameterization for the domain of the adjoint operator. This direct decomposition is written in terms of the semi-deficiency subspaces and is an analogue of the von Neumann formulas (1.7) and (1.13) for the case of the symmetric operator ?A whose domain is not dense in H.