Fusion Frames for Operators in Hilbert C *-Modules
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DOI: 10.1007/s13226-020-0432-6
FUSION FRAMES FOR OPERATORS IN HILBERT C ∗ -MODULES M. Khayyami∗ and A. Nazari∗∗ ∗ Department
of Mathematics, Kerman Branch, Islamic Azad University, Kerman, Iran
∗∗ Department
of Pure Mathematics Faculty of Mathematics and Computer,
Shahid Bahonar University of Kerman, Kerman, Iran 76169-14111 e-mails: [email protected]; [email protected] (Received 16 February 2018; after final revision 31 December 2018; accepted 19 March 2019) In this paper we introduce K -fusion frames on a Hilbert C ∗ -module H, where K is an adjointable operator on H. We obtain several characterizations of K -fusion frames. In addition, we extend the concept of duality to K -fusion frames and study some of its properties. Key words : Frame; fusion frame; operator; duality; Hilbert C ∗ -module. 2010 Mathematics Subject Classification : 42C15, 46L05.
1. I NTRODUCTION The concept of frame in Hilbert spaces has been introduced by Duffin and Schaeffer [3] in 1952 to study some deep problems in nonharmonic Fourier series. After it has been reintroduced and developed in 1986 by Daubechies, Grossmann and Meyer [2], frame theory began to be widely used, particularly in the more specialized context of wavelet frames and Gabor frames. Traditionally, frames have been used in signal processing, image processing, data compression and sampling theory. In 2000, Frank-Larson [6] introduced the notion of frames in Hilbert C ∗ -modules as a generalization of frames in Hilbert spaces, and Jing [10] continued to consider them. It is well known that Hilbert C ∗ -modules are generalizations of Hilbert spaces by allowing the inner product to take values in a C ∗ -algebra rather than in the field of complex numbers. Recentely, Khosravi and Khosravi [11] introduced the fusion frame theory in Hilbert C ∗ -modules.
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M. KHAYYAMI AND A. NAZARI Atomic systems for subspaces were first introduced by Feichtinger and Werther [5] based on
examples arising in sampling theory. In [7], Gavruta introduced atomic systems for operators in Hilbert spaces, and frames for operators allowing the reconstruction of elements from the range of a linear and bounded operator. Recently, Najati [13] generalized the notion of frames for operators for Hilbert spaces to Hilbert C ∗ -modules and studied som of their properties. In this paper we define the concept of fusion frames for operators in Hilbert C ∗ -modules. The paper is organized in the following manner. In Section 2, we recall the definitions and basic properties. Section 3 is devoted to introduce fusion frames for operators in Hilbert C ∗ -modules. In Section 4, we study the concept of duality of fusion frames for operators in Hilbert C ∗ -modules based on the notion of duality of fusion frames introduced in [8, 9]. 2. P RELIMINARIES In the following we briefly recall some definitions and basic properties of operators and fusion frames in Hilbert C ∗ -modules. Throughout this paper, the symbols I, C and A refer, respectively to a finite or countable index set, the field of complex numbers
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