Generalized resolvents of an isometric operator in a Pontrjagin space (Q1071284)

The operator V in the Pontrjagin space \({\mathfrak H}\) is said to be contractive if [Vf\(| Vf]\leq [f| f]\) for all \(f\in {\mathfrak D}(V)\). An operator U, acting in a \(\Pi_{\kappa}\)-space \({\mathfrak K}\) extending \({\mathfrak H}\) is named a dilation of V if \(V=PU|_{{\mathfrak H}}\), where P is the orthogonal projection of \({\mathfrak K}\) onto \({\mathfrak H}\). If V is a closed injective isometric operator in \({\mathfrak H}\) and the bounded operator W in \({\mathfrak H}\) is a contractive extension of V, then the mapping \(z\mapsto (I-zW)^{-1},\) \(z^{-1}\in \rho (W)\), is called a contractive resolvent of V. If the unitary operator U, acting in \({\mathfrak K}\), is an extension of V, then the mapping \(z\mapsto R(z):=P(I-zU)^{- 1}|_{{\mathfrak H}}\) defines the generalized resolvent of V. Using the canonical decompositions of \({\mathfrak H}\) determined by \({\mathfrak D}(V)\) and \({\mathfrak R}(V)\), the author proves that a mapping R is a regular generalized resolvent of a closed injective isometric operator V if and only if it has the representation \(R(z)=\{I-z[V+\Phi (z)]\}^{- 1}\) for almost all \(z\in {\mathbb{C}}_ 0\), where \(\Phi\) is meromorphic in \({\mathbb{C}}_ 0\). The main theorems express bijective correspondences between the set of all contractive (regular generalized) resolvents R of V and the set \({\mathcal E}_ 0\) of all \(\Gamma\)-accretive linear relations of a Hilbert space \({\mathfrak G}\) (respectively a set of mappings from \({\mathbb{C}}_ 0\) into \({\mathcal E}_ 0)\) via the formula \(R(z)=(I-zU)^{- 1}+\Gamma_{1/z}\{\Theta (z)+E(z)\}^{-1}\Gamma^+_{\bar z}.\)