Frames for Operators in Hilbert and Banach Spaces

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.