Generalized frames for operators associated with atomic systems. (English) Zbl 1382.42019

Li, Dongwei; Leng, Jinsong; Huang, Tingzhu

Banach J. Math. Anal. 12, No. 1, 206-221 (2018).

Summary: In this paper, we investigate the g-frame and the Bessel g-sequence related to a linear bounded operator \(K\) in a Hilbert space, which we call a \(K\)-g-frame and a \(K\)-dual Bessel g-sequence, respectively. Since the frame operator for a \(K\)-g-frame may not be invertible, there is no classical canonical dual for a \(K\)-g-frame. So we characterize the concept of a canonical \(K\)-dual Bessel g-sequence of a \(K\)-g-frame that generalizes the classical dual of a g-frame. Moreover, we use a family of linear operators to characterize atomic systems. We also consider the construction of new atomic systems from given ones and bounded operators.

MSC:

42C15	General harmonic expansions, frames
42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems
47B32	Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)

Keywords:

\(K\)-g-frames; atomic system; \(K\)-dual Bessel g-sequence; canonical \(K\)-dual Bessel g-sequence