Completion, extension, factorization, and lifting of operators with a negative index

The famous results of M.G. Kre\u{\i}n concerning the description of selfadjoint contractive extensions of a Hermitian contraction $T_1$ and the characterization of all nonnegative selfadjoint extensions $\widetilde A$ of a nonnegative operator $A$ via the inequalities $A_K\leq \widetilde A \leq A_F$, where $A_K$ and $A_F$ are the Kre\u{\i}n-von Neumann extension and the Friedrichs extension of $A$, are generalized to the situation, where $\widetilde A$ 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 $2\times 2$ nonnegative block operators. The extension of this fundamental result allows us to prove analogs of the above mentioned results of M.G. Kre\u{\i}n and, in addition, to solve some related lifting problems for $J$-contractive operators in Hilbert, Pontryagin and Kre\u{\i}n 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.