On uncountable b-frames in non-separable Hilbert spaces

Abstract

This work is dedicated to uncountable frames in non-separable Hilbert spaces associated with bilinear mappings. Bounded bilinear mapping is considered, and using this mapping, the concepts of uncountable b-Besselian system, b-frame and b-frame operator are introduced. Criteria for uncountable b-Besselness and b-frameness of system are proved, and some properties of uncountable b-frame operator are established. Stability and perturbation of uncountable b-frames in non-separable Hilbert spaces are studied. From the obtained results, in particular, corresponding results for tensors are derived.Keywords. non-separable Hilbert space • uncountable frame • frame operator • Noetherian perturbation