An Appropriate Representation Space for Controlled g-Frames
- PDF / 304,550 Bytes
- 14 Pages / 439.37 x 666.142 pts Page_size
- 39 Downloads / 142 Views
An Appropriate Representation Space for Controlled g-Frames Maryam Forughi1 · Elnaz Osgooei2
· Asghar Rahimi3 · Mojgan Javahernia1
Received: 2 October 2019 / Revised: 13 April 2020 / Accepted: 12 May 2020 © Iranian Mathematical Society 2020
Abstract In this paper, motivating the range of operators, we propose an appropriate representation space to introduce synthesis and analysis operators of controlled g-frames and discuss the properties of these operators. Especially, we show that the operator obtained by the composition of the synthesis and analysis operators of two controlled g-Bessel sequence is a trace class operator. Also, we define the canonical controlled g-dual and show that this dual gives rise to expand coefficients with the minimal norm. Finally, we extend some known equalities and inequalities for controlled g-frames. Keywords Controlled g-frame · Controlled g-dual frame · Trace class operator Mathematics Subject Classification 42C15 · 42C40 · 41A58
1 Introduction and Preliminaries Frames were first introduced in the context of non-harmonic Fourier series by Duffin and Schaeffer [6]. During the last 20 years, the theory of frames has been developed
Communicated by Fereshteh Sady.
B
Elnaz Osgooei [email protected] Maryam Forughi [email protected] Asghar Rahimi [email protected] Mojgan Javahernia [email protected]
1
Department of Mathematics, Shabestar Branch, Islamic Azad University, Shabester, Iran
2
Faculty of Science, Urmia University of Technology, Urmia, Iran
3
Department of Mathematics, University of Maragheh, Maragheh, Iran
123
Bulletin of the Iranian Mathematical Society
rapidly„ and because of the abundant use of frames in engineering and applied sciences, many generalization of frames have come into play. g-frames that include the concept of ordinary frames have been introduced by Sun [15] and improved by many authors [5,8,10,14]. Controlled frames have been improved recently to improve the numerical efficiency of interactive algorithms that inverts the frame operator [2]. Following that, controlled frames have been generalized to another kinds of frames [5,7–9,12–14]. In this paper, motivating the concept of g-frames and controlled frames, we define controlled g-frames. In Sect. 2, a new representation space is introduced, such that the synthesis and analysis operators could be defined. In Sect. 3, controlled g-dual frames and canonical controlled g-dual frames are introduced and shown that canonical g-dual gives rise to expand coefficients with the minimal norm. Finally, some equalities and inequalities are presented for controlled g-frames and especially for their operators in Sect. 4. Throughout this paper, H is a separable Hilbert space, {Hi }i∈I is the collection of Hilbert spaces, B(H , K ) is the family of all linear bounded operators from H into K and GL(H ) is the set of all bounded linear operators which have bounded inverses. At first, we collect some definitions and basic results that are needed in the paper. Lemma 1.1 [11] Let u ∈ B(H ) be a
Data Loading...
