Frames for Krein spaces

JUAN IGNACIO GIRIBET; ALEJANDRA MAESTRIPIERI; FRANCISCO MARTÍNEZ PERÍA; PEDRO MASSEY

Sevilla

Workshop; XXII International Workshop in Operator Theory and Applications; 2011

Given a (complex) Hilbert space H, a family of vectors {fi}iÎI in H is a frame for H if there exist constants 0< A B such that A ||f||2 £ S iÎI ||2 £ B ||f||2. Notice that every frame for H is a generating system for H. Some obvious examples of frames for a Hilbert space are its orthogonal, orthonormal and Riesz bases. However, if {fi}iÎI is a frame for H and {}iÎI are the frame coefficients of a vector fÎH, it is possible to reconstruct f faithfully, even if some of the coefficients are missing. In fact, the data redundancy obtained after analyzing a vector (signal) with a frame, made them useful in engineering applications such as signal processing. On the other hand, given a Krein space K with fundamental symmetry J, the notion of J-orthonormalized system (and basis) is linked with the existence of a (maximal) dual pair. Recall that a pair of subspaces (L+, L-) in K is a dual pair if L+ is J-nonnegative, L- is J-nonpositive and they are J-orthogonal. The purpose of this talk is to study a particular class of frames for a fixed a Krein space K, hereafter mentioned as J-frames. To each J-frame for K there is associated a pair of uniformly J-definite maximal subspaces, but they are not necessarily J-orthogonal. Some characterizations are discussed and some basic frame theory problems are extrapolated to J-frames. The talk is based on a joint work with J. Giribet, A. Maestripieri and P. Massey.