Toeplitz $O$-Frames for Operators in Banach Spaces

Abstract

We define Toeplitz $O$-frame for operators as a generalization of the notion of $O$-frame introduced by Reinov [11]. A necessary condition for the existence of a Toeplitz $O$-frame is given. It has been proved that an $O$-frame for operators can generate a Toeplitz $O$-frame from a given Toeplitz matrix but the converse need not be true. Also, a sufficient condition on infinite matrices for the existence of an $O$-frame is given. Finally, the notion of a strong $O$-frame is defined and a necessary and sufficient condition for its existence has been obtained.