On frames for Krein spaces

GIRIBET, J.I; MAESTRIPIERI, A.; MARTÍNEZ PERÍA, FRANCISCO D; MASSEY, PEDRO

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2012 vol. 393 p. 122 - 137

0022-247X

A definition of frames for Krein spaces is proposed, which extends the notion of $J$-orthonormal bases 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, 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.