Book Description
Let [italic]H be a Hilbert space. Formally normal, normal, symmetric, selfadjoint, and semibounded subspaces of [italic]H2=[italic]H2[circled plus][italic]H are defined by means of the corresponding properties of the graphs of operators in H which are formally normal, normal, symmetric, selfadjoint, or semibounded, respectively. The author gives a complete description of all formally normal and normal subspace extensions in [italic]H2 of a given formally normal subspace [italic]N of [italic]H2. Those extensions which are graphs of operators are explicitly characterized. The symmetric and selfadjoint extensions of a given symmetric subspace are also classified; this result extends the well-known result of von Neumann characterizing the selfadjoint extensions of a (densely defined) symmetric operator. The construction of the "Friedrichs extension'' of a semibounded symmetric subspace is outlined. The existence of formally normal and symmetric extensions in a larger Hilbert space is also studied. A formally normal subspace need not have any normal subspace extension in a bigger subspace. But (as is known for operators), every symmetric subspace has selfadjoint extensions in suitable larger spaces; these extensions are completely characterized.
