On G−Banach Frames

Abstract

Abdollahpour et. al [1] generalized the concepts of frames for Banach spaces and defined g-Banach frames in Banach spaces. In the present paper, we define various types of g-Banach frames in Banach spaces. Examples and counter examples to distinguish various types of g-Banach frames in Banach spaces have been given. It has been proved that if a Banach space X has a Banach frame, then X has a normalized tight g-Banach frame for X. A characterization of an exact g-Banach frame has been given. Also, we consider the finite sum of g-Banach frames and give a sufficient condition for the finite sum of g-Banach frames to be a g-Banach frame. Finally, a sufficient condition for the stability of g-Banach frames in Banach spaces which provides optimal frame bounds has been given.

Key words and phrases. Frame, Banach frame, g-frame, g-Banach frame, stability.

2000 Mathematics Subject Classification. 42A38, 42C15, 46B15, 41A58