Atomic Subspaces for Operators
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DOI: 10.1007/s13226-020-0448-y
ATOMIC SUBSPACES FOR OPERATORS1 Animesh Bhandari and Saikat Mukherjee Department of Mathematics, NIT Meghalaya, Shillong 793 003, India e-mails: [email protected]; [email protected] (Received 10 January 2018; after final revision 27 March 2019; accepted 22 May 2019) This paper introduces the concept of atomic subspaces with respect to a bounded linear operator. Atomic subspaces generalize fusion frames and this generalization leads to the notion of K-fusion frames. Characterizations of K-fusion frames are discussed. Various properties of K-fusion frames, for example, direct sum, intersection, are studied. Key words : Atomic subspaces; frames; K-fusion frames. 2010 Mathematics Subject Classification : 42C15, 46C15.
1. I NTRODUCTION Notion of Hilbert space frames was first introduced by Duffin and Schaeffer [6] in 1952 to reconstruct signals. Much later in the year 1986, the fundamental concept of frames and their significance in signal processing, image processing and data processing were presented by Daubechies, Grossman and Meyer [4]. Frame theory plays an important role in various fields and have been widely applied in signal processing [8], sampling theory [7], coding and communications [11, 15] and so on. It is a well-known fact that every element in a separable Hilbert space H can be explicitly represented as a linear combination of an orthonormal basis in H with the help of Fourier coefficients. But if one of the basis elements, for some reason, is removed, the explicit representation may not hold. Primarily due to this reason an overcomplete system was introduced which satisfies the explicit representation but more flexible when f ∈ H is to be reconstructed. Such an overcomplete system is called a “Frame”. 1
Second author is supported by NIT Meghalaya Start-up Grant Project and DTS-SERB project MTR/2017/000797.
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ANIMESH BHANDARI AND SAIKAT MUKHERJEE
L. Gˇavrut¸a in [9] was first to introduce the notion of K-frames to study the nature of atomic systems for a separable Hilbert space H with respect to a bounded linear operator K on H. In [10], Gˇavrut¸a further studied atomic systems for operators in reproducing kernel Hilbert spaces, especially Bergman and Fock spaces. It is well-known fact that K-frames are more general than the classical frames and due to higher generality of K-frames, many properties of frames may not hold for Kframes. In the 21st century scientists introduced fusion frames to handle massive amount of data to obtain mathematical framework to model and analyze such problems, which are otherwise almost impossible to handle. Moreover fusion frames are also significantly important mathematical gadget for theory oriented mathematical problems in frame theory. The notion of fusion frames (or frames of subspaces) was first introduced by Casazza et al. (see [1, 2]). There are so many applications of fusion frames like coding theory, compressed sensing, data processing and so on. A fusion frame is a frame-like collection of closed subspaces in a Hi
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