Symmetric extensions of symmetric linear relations (operators) preserving the multivalued part

Abstract

Let $\mathfrak H$ be a Hilbert space and let $A$ be a symmetric linear relation (in particular, a nondensely defined operator) in $\mathfrak H$. By using the concept of a boundary triplet for $A^*$ we characterize symmetric extensions $\widetilde A\supset A$ preserving the multivalued part of $A$. Such a characterization is given in terms of an abstract boundary parameter and the Weyl function of the boundary triplet. Application of these results to the Hamiltonian system $Jy'-B(t)y=\lambda\Delta(t) y$ enabled us to describe its matrix solutions generating the generalized Fourier transform with the nonempty set of respective spectral functions.