The famous results of M.G. Kre\u{\i}n concerning the description of
selfadjoint contractive extensions of a Hermitian contraction T1β and the
characterization of all nonnegative selfadjoint extensions A of a
nonnegative operator A via the inequalities AKββ€Aβ€AFβ,
where AKβ and AFβ are the Kre\u{\i}n-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 IβT1ββT1β and A, respectively;
these conditions are automatically satisfied if T1β 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
2Γ2 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.Comment: 29 page