New characterizations of \(g\)-Bessel sequences and \(g\)-Riesz bases in Hilbert spaces (Q889136): Difference between revisions

The author studies properties of g-Bessel sequences, g-Riesz bases and g-orthonormal bases in Hilbert spaces, establishing several properties. In Section 2 the author shows some properties about g-Bessel sequences. There are, among others: Theorem 2.16 characterizes a linear bounded operator from \(l^2(\oplus H_j)\) into \(H\); Theorem 2.18 characterizes the g-Bessel sequence if the bound is not involved; Theorem 2.19 gives a necessary and sufficient condition for it to be a g-frame in terms of its analysis operator; Theorem 2.21 relates g-Bessel sequences with sequences satisfying a lower g-frame condition and g-Riesz Fischer sequences. In Section 3 the author shows the main results of this paper. Theorem 3.4 characterizes g-Riesz bases in terms of the uniqueness of dual g-frames. Theorem 3.6 establishes the uniqueness of a normalized tight dual g-frame for a normalized tight g-frame. Finally, Theorems 3.7 and 3.8 characterize g-orthonormal bases.