Frames for Operators in Banach Spaces
- PDF / 241,716 Bytes
- 9 Pages / 439.642 x 666.49 pts Page_size
- 53 Downloads / 265 Views
Frames for Operators in Banach Spaces Ramu Geddavalasa1 · P. Sam Johnson1
Received: 4 December 2016 / Accepted: 29 March 2017 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2017
Abstract 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. Keywords Xd -atomic system · Xd -K-frame Mathematics Subject Classification (2010) 47B32 · 42C15
1 Introduction Frames are a tool for the construction of series expansions in Hilbert spaces. Frames provide stable expansions, quite in contrast to orthogonal expansions — they may be overcomplete and the coefficients in the frame expansion therefore need not be unique. The redundancy
Ramu Geddavalasa
[email protected] P. Sam Johnson [email protected] 1
Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Surathkal, Mangaluru 575 025, India
R. Geddavalasa, P. S. Johnson
and flexibility offered by frames has spurred their applications in a variety of areas throughout mathematics and engineering, such as operator theory [11], harmonic analysis [9], pseudo-differential operators [10], quantum computing [5], signal and image processing [4], and wireless communication [13]. Theoretical research of frames for Banach spaces is quite different from that for Hilbert spaces. Due to the lack of an inner product, the properties of Hilbert frames usually do not transfer automatically to Banach spaces. Gr¨ochenig [8] generalized Banach frames with respect to certain sequence spaces. The main feature of frames that Gr¨ochenig was trying to capture in a general Banach space was the unique association of a vector in a Hilbert space with the natural set of frame coefficients. After the work of Gr¨ochenig, frames in Banach spaces have become topic of investigation for many mathematicians. A sequence space Xd is called a BK-space if it is a Banach space and the coordinate functionals are continuous on Xd . If the canonical unit vectors form a Schauder basis for Xd , then Xd is called a CB-space and its canonical basis is denoted by {en }. If Xd is reflexive and a CB-space, then Xd is called an RCB-space. Also, the dual of Xd is denoted by Xd∗ . When Xd∗ is a CB-space, then its canonical basis is denoted by {en∗ }. We
Data Loading...
