On selfadjoint dilation of the dissipative extension of a direct sum differential operator

Abstract

In this paper, we describe all maximal dissipative, maximal accretive and selfadjoint extensions of the minimal symmetric direct sum differential operators. Further using the equivalence of the Lax-Phillips scattering function and the Sz.-Nagy-Foiaş characteristic function we show that all eigen and associated functions of the maximal dissipative extension of the minimal symmetric direct sum operator are complete in $L_{w}^{2}(\Omega ),$ where $\Omega =\Omega _{1}\cup \Omega _{2},$ $\Omega _{1}=(0,c)$ and $\Omega _{2}=(c,\infty ).$