Extension theory for operators and spaces with indefinite metric

We show that the classical extension theory for isometric operators cannot be automatically extended to $J$-isometric and $J$-Hermitian operators in $J$-spaces with infinite rank. We construct a single extension theory which includes both the isometric and Hermitian operators in a Hilbert space and the $J$-isometric and $J$-Hermitian operators in a $J$-space with arbitrary indefinite rank. The basis of the construction is a scheme for extension of a neutral subspace of a $J$-space to a maximal or hypermaximal subspace.