Frame Operators for Frames in Krein Spaces

In recent years, frames in Krein spaces and several generalizations have been extensively studied. In this paper, we propose an alternative way of looking at the notion of frames in Krein spaces and give a necessary and sufficient condition for a sequence in a Krein space to be a Bessel sequence. We observe that a subsequence of a frame in a Krein space need not be a frame. Also, two complementary subsequences are considered in which one of them is a frame for a Krein space. We obtain necessary and sufficient conditions under which the other one is also a frame for the Krein space.