The famous results of M.G. Kreȋn concerning the description of selfadjoint contractive extensions of a Hermitian contraction T 1 and the characterization of all nonnegative selfadjoint extensions A of a nonnegative operator A via the inequalities A K ≤ A ≤ A F , where A K and A F are the Kreȋn-von Neumann extension and the Friedrichs extension of A, are generalized to the situation, where A is allowed to have a fixed number of negative eigenvalues. These generalizations are shown to be possible under a certain minimality condition on the negative index of the operators I − T * 1 T 1 and A, respectively; these conditions are automatically satisfied if T 1 is contractive or A is nonnegative, respectively.The approach developed in this paper starts by establishing first a generalization of an old result due to Yu.L. Shmul'yan on completions of 2 × 2 nonnegative block operators. The extension of this fundamental result allows us to prove analogs of the above mentioned results of M.G. Kreȋn and, in addition, to solve some related lifting problems for J-contractive operators in Hilbert, Pontryagin and Kreȋn spaces in a simple manner. Also some new factorization results are derived, for instance, a generalization of the well-known Douglas factorization of Hilbert space operators. In the final steps of the treatment some very recent results concerning inequalities between semibounded selfadjoint relations and their inverses turn out to be central in order to treat the ordering of non-contractive selfadjoint operators under Cayley transforms properly.

An isometric operator V in a Pontryagin space H is called standard, if its domain and the range are nondegenerate subspaces in H. Generalized resolvents of standard isometric operators were described in [11]. In the present paper generalized resolvents of non-standard Pontryagin space isometric operators are described. The method of the proof is based on the notion of boundary triplet of isometric operators in Pontryagin spaces. In the Hilbert space setting the notion of boundary triplet for isometric operators was introduced in [17].

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A generalization of the well-known results of M.G. Kreȋn about the description of selfadjoint contractive extension of a hermitian contraction is obtained. This generalization concerns the situation, where the selfadjoint operator A and extensions A belong to a Kreȋn space or a Pontryagin space and their defect operators are allowed to have a fixed number of negative eigenvalues. Also a result of Yu.L. Shmul'yan on completions of nonnegative block operators is generalized for block operators with a fixed number of negative eigenvalues in a Kreȋn space.

An indefinite variant of the abstract interpolation problem is considered. Associated to this problem is a model Pontryagin space isometric operator V . All the solutions of the problem are shown to be in a one-to-one correspondence with a subset of the set of all unitary extenions U of V . These unitary extension U of V are realized as unitary colligations with the indefinite de Branges-Rovnyak space D(s) as a state space.