Infinite Matrix Operators and $E$-Frames in Hilbert Spaces

Abstract

The concept of an $E$-frame, recently introduced in frame theory, is obtained by applying an infinite invertible complex matrix to a sequence of elements of a Hilbert space $\mathcal{H}$. Here, $E$ is considered as a matrix mapping on the sequence space ${⨁}_{n=1}^{\infty }\mathcal{H}$. A natural question is to determine the conditions on a sequence $\Psi =\{{\psi }_k{\}}_{k=1}^{\infty }$ and a matrix $E$ under which $E\Psi$ becomes a frame. In this paper, we show that if $E$ acts surjectively on ${⨁}_{n=1}^{\infty }\mathcal{H}$, then every ordinary frame is an $E$-frame.