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A generalization of the well-known results of M.G. Kre\u{\i}n about thedescription of selfadjoint contractive extension of a hermitian contraction isobtained. This generalization concerns the situation, where the selfadjointoperator $A$ and extensions $\widetilde A$ belong to a Kre\u{\i}n space or aPontryagin space and their defect operators are allowed to have a fixed numberof negative eigenvalues. Also a result of Yu.L. Shmul'yan on completions ofnonnegative block operators is generalized for block operators with a fixednumber of negative eigenvalues in a Kre\u{\i}n space.  This paper is a natural continuation of S. Hassi's and author's paper [5].

Completion and extension of operators in Kre\u{\i}n spaces