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. 
J. I. Giribet, A. Maestripieri, F. Mart\'inez Per\'ia and P. Massey
math.FA