On $J$-frames related to maximal definite subspaces

A definition of frames in Krein spaces is proposed which extends the concept of $J$-frames defined by J.I. Giribet et al., J. Math. Anal. Appl. $\mathbf{\text{393}}$ (2012), 122-137. The principal difference consists in the fact that a $J$-frame is related to maximal definite subspaces ${\mathcal{M}}_{\pm }$ which are not assumed to be uniformly definite. The latter allows one to extend the collection of $J$-frames. In particular, some complete $J$-orthogonal sequences and $J$-orthogonal Schauder bases can be interpreted as $J$-frames.