K–g-fusion frame generators for unitary systems on Hilbert spaces

Abstract

K–g-fusion frame is a generalization of several currently popular frames, including the fusion frames, K-fusion frames and g-fusion frames. And structured frames are one of most interest current research subjects. In this paper, we mainly study structured K–g-fusion frames, i.e. K–g-fusion frames generated by an operator-space pair through a unitary system. We call such pair a K–g-fusion frame generator of the unitary system. We study some properties of K–g-fusion frame generators, which shows that under some condition we can obtain K–g-fusion frame generators by some simple ways. Moreover, under some condition we characterize K–g-fusion frame generators in terms of complete wandering subspaces and generalized local commutants. Finally, we discuss the duals of a K–g-fusion frame generator, which is called K–g-fusion dual Bessel generator of the generator. And under some weak condition, we completely characterize the structure of all K–g-fusion dual Bessel generators for a Parseval K–g-fusion frame generator and give an explicit construction of the canonical K–g-fusion dual Bessel generator for a Parseval K–g-fusion frame generator. Moreover, we show that under certain condition, existence of dual on range of K guarantees a K–g-fusion Bessel generator to be a K–g-fusion frame generator, and we also can generate a K–g-fusion frame generator from a K–g-fusion dual Bessel generator.