Frames for Operators in Banach Spaces
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A family of local atoms in a Banach space has been introduced and it has been generalized to an atomic system for operators in Banach spaces, which has been further led to introduce new frames for operators by Dastourian and Janfada, by making use of semi-inner products. Unlike the traditional way of considering sequences in the dual space, sequences in the original space are considered to study them. Appropriate changes have been made in the definitions of atomic systems and frames for operators to fit them for sequences in the dual space without using semi-inner products so that the new notion for Banach spaces can be thought of as a generalization of Banach frames. With some crucial assumptions, we show that frames for operators in Banach spaces share nice properties of frames for operators in Hilbert spaces.
KeywordsXd-atomic system Xd-K-frame
Mathematics Subject Classification (2010)47B32 42C15
The present work of second author was partially supported by National Board for Higher Mathematics (NBHM), Ministry of Atomic Energy, Government of India (Reference No.2/48(16)/2012/ NBHM(R.P.)/R&D 11 /9133) and the first author thanks the National Institute of Technology Karnataka (NITK), Surathkal for giving him financial support.