Generalized Resolvents of Dual Pairs of Contractions

Let H be a Hilbert space. The contractions T, S in H with closed, nondense domains D (T), D (S) are said to form a dual pair (denoted by {T,S}), if the relation $$(Tx,y)=(x,Sy):\ (x\varepsilon \mathit{D}(T)),:\ y\varepsilon \mathit{D}(S))$$ holds. It is well-known that for each dual pair {T,S} of contractions in H there exists a contraction T~ in H, D (T~)=H, such that $$T\subset \check{T},:\ S\subset \check{T}^{*}$$ (1) .