On a family of frames for Krein spaces

A definition of frames for Krein spaces is proposed, which extends the notion of $J$-orthonormal basis of Krein spaces. A $J$-frame for a Krein space $(HH, K{,}{,})$ is in particular a frame for $HH$ in the Hilbert space sense. But it is also compatible with the indefinite inner product $K{,}{,}$, meaning that it determines a pair of maximal uniformly $J$-definite subspaces with different positivity, an analogue to the maximal dual pair associated to a $J$-orthonormal basis. Also, each $J$-frame induces an indefinite reconstruction formula for the vectors in $HH$, which resembles the one given by a $J$-orthonormal basis.