Frames for Operators in Hilbert and Banach Spaces
Date
2017
Authors
Geddavalasa, Ramu
Journal Title
Journal ISSN
Volume Title
Publisher
National Institute of Technology Karnataka, Surathkal
Abstract
The notion of K-frames has been introduced by Laura G˘avrut¸a to study the
atomic systems with respect to a bounded linear operator K in a separable Hilbert
space. K-frames are more general than ordinary frames in the sense that the
lower frame bound only holds for the elements in the range of K. Because of
the higher generality of K-frames, many properties for ordinary frames may not
hold for K-frames, such as the corresponding synthesis operator for K-frames is
not surjective, the frame operator for K-frames is not isomorphic, the alternate
dual reconstruction pair for K-frames is not interchangeable in general. Note that
the frame operator S for a K-frame is semidefinite, so there is also S1=2, but not
positive. Operators that preserve K-frames and generating new K-frames from
old ones by taking sums have been discussed. A close relation between K-frames
and quotient operators is established using through operator-theoretic results on
quotient operators and few characterizations are given.
A frame for a Banach space X was defined as a sequence of elements in X ∗,
not of elements in the original space X . However, semi-inner products for Banach spaces make possible the development of inner product type arguments in
Banach spaces. The concept of a family of local atoms in a Banach space X with
respect to a BK-space Xd was introduced by Dastourian and Janfada using a semiinner product. This concept was generalized to an atomic system for an operator
K 2 B(X ) called Xd∗-atomic system and it has been led to the definition of a new
frame with respect to the operator K, called Xd∗-K-frame. Appropriate changes
have been made in the definitions of X ∗
d -atomic systems and Xd∗-K-frames to fit
them for sequences in the dual space without using semi-inner products, called
Xd-atomic systems and Xd-K-frames respectively. New Xd-K-frames are generated from each Xd-frame for a Banach space X and each operator K 2 B(X ∗) and
some characterizations are given. With some crucial assumptions, it is shown that
frames for operators in Banach spaces share nice properties of frames for operators
in Hilbert spaces.
Description
Keywords
Department of Mathematical and Computational Sciences, Frame, K-frame, Xd-atomic system, Xd-K-frame