Some Results on Generalized Frames

Javad Baradaran, Zahra Ghorbani

Abstract

The concept of a generalized frame or simply a $g$-frame in a Hilbert space $H$ was introduced by Wenchang Sun in [4]. Given a $g$-frame $\{\Lambda_i\}_{i \in I}$ in a Hilbert space $H$ and a bounded operator $T$ on $H$, we show that the sequence $\{\Lambda_{i}T\}_{i \in I}$ is a $g$-frame for $H$ if and only if $T$ is invertiable on $H$. Moreover, we prove that add a $g$-frame to its canonical dual $g$-frame and the canonical Parseval $g$-frame are also $g$-frames. At the end, we provide sufficient conditions under which a subsequence of a $g$-frame in a Hilbert space $H$ is itself a $g$-frame for $H$.