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The famous results of M.G. Kre\u{\i}n concerning the description ofselfadjoint contractive extensions of a Hermitian contraction $T_1$ and thecharacterization of all nonnegative selfadjoint extensions $\widetilde A$ of anonnegative 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 theFriedrichs extension of $A$, are generalized to the situation, where$\widetilde A$ is allowed to have a fixed number of negative eigenvalues. Thesegeneralizations are shown to be possible under a certain minimality conditionon 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 $A$ isnonnegative, respectively.  The approach developed in this paper starts by establishing first ageneralization of an old result due to Yu.L. Shmul'yan on completions of$2\times 2$ nonnegative block operators. The extension of this fundamentalresult 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 asimple manner. Also some new factorization results are derived, for instance, ageneralization of the well-known Douglas factorization of Hilbert spaceoperators. In the final steps of the treatment some very recent resultsconcerning inequalities between semibounded selfadjoint relations and theirinverses turn out to be central in order to treat the ordering ofnon-contractive selfadjoint operators under Cayley transforms properly.

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