On $\mathbf{\mathbf{E}}$-Frames in separable Hilbert Spaces

Abstract

The purpose of this paper is to introduce the concept of $\mathrm{E}-$frames for a separable Hilbert space ${\mathcal H},$ where $\mathrm{E}$ is an invertible infinite matrix mapping on the Hilbert space $\mathop \oplus \limits_{n = 1}^\infty {{ \mathcal H}}$. We investigate and study some properties of $\mathrm{E}-$frames and characterize all $\mathrm{E}-$frames for ${\mathcal H}$. Further more, we characterize all dual $\mathrm{E}-$frames associated with a given $\mathrm{E}-$frame. A similar characterization is also established for $\mathrm{E}-$orthonormal bases, $\mathrm{E}-$Riesz bases and dual $\mathrm{E}-$Riesz bases. In continue we obtain a lower estimate for the lower bound of some matrix operators on the $p-$bounded variation sequence space $bv_p$ and Euler weighted sequence space $e_{w,p}^\theta$. Then we deal with several types of $\mathrm{E}-$frames such as $\mathbf{\Delta}-$frames and Euler frames for ${\mathcal H}$ which are related to the Hilbert spaces $bv_2$ and $e_{2}^\theta$, respectively.

Key-Words: E-frame; E-orthonormal basis; E-Bessel sequence; E-Riesz basis; Direct sum of Hilbert spaces; Euler frame; $\Delta-$frame.