g-FRAMES AND MODULAR RIESZ BASES IN HILBERT C*-MODULES

2012 ◽  
Vol 10 (02) ◽  
pp. 1250013 ◽  
Author(s):  
AMIR KHOSRAVI ◽  
BEHROOZ KHOSRAVI
Keyword(s):  
Hilbert Spaces ◽  
Riesz Basis ◽  
Riesz Bases ◽  
Parseval Frame ◽  
C Algebra ◽  
Perturbation Result ◽  
Natural Way

In this paper we introduce modular Riesz basis, modular g-Riesz basis in Hilbert C*-modules in a very natural way and we show that they share many properties with Riesz basis and g-Riesz basis in Hilbert spaces. We also found that by using the fact that every finitely or countably generated Hilbert C*-module over a unital C*-algebra has a standard Parseval frame, we characterize g-frames, modular Riesz bases and modular g-Riesz bases. Finally we obtain a perturbation result for modular g-Riesz bases.

C∗-algebra consisting of all g-Bessel sequences in a Hilbert space

2017 ◽  
Vol 15 (01) ◽  
pp. 1750004
Author(s):  
Z. L. Chen ◽  
H. X. Cao ◽  
Z. H. Guo
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Small Perturbation ◽  
Hilbert Spaces ◽  
Operator Theory ◽  
Operator Space ◽  
Riesz Bases ◽  
C Algebra ◽  
Linear Bijection ◽  
Bessel Sequences

For Hilbert spaces [Formula: see text] and [Formula: see text], we use the notations [Formula: see text], [Formula: see text] and [Formula: see text] to denote the sets of all [Formula: see text]-Bessel sequences, [Formula: see text]-frames and Riesz bases in [Formula: see text] with respect to [Formula: see text], respectively. By defining a linear operation and a norm, we prove that [Formula: see text] becomes a Banach space and is isometrically isomorphic to the operator space [Formula: see text], where [Formula: see text]. In light of operator theory, it is proved that [Formula: see text] and [Formula: see text] are open sets in [Formula: see text]. This implies that both [Formula: see text]-frames and Riesz bases are stable under a small perturbation. By introducing a linear bijection [Formula: see text] from [Formula: see text] onto the [Formula: see text]-algebra [Formula: see text], a multiplication and an involution on the Banach space [Formula: see text] are defined so that [Formula: see text] becomes a unital [Formula: see text]-algebra that is isometrically isomorphic to the [Formula: see text]-algebra [Formula: see text], provided that [Formula: see text] and [Formula: see text] are isomorphic.

scholarly journals Gram matrix associated to controlled frames

2018 ◽  
Vol 16 (05) ◽  
pp. 1850035 ◽  
Author(s):  
E. Osgooei ◽  
A. Rahimi
Keyword(s):  
Hilbert Spaces ◽  
Diagnostic Tool ◽  
Riesz Basis ◽  
Trace Class ◽  
Trace Class Operator ◽  
Riesz Bases ◽  
Gram Matrix ◽  
Frame Operator ◽  
Class Operator ◽  
Interactive Algorithms

Controlled frames have been recently introduced in Hilbert spaces to improve the numerical efficiency of interactive algorithms for inverting the frame operator. In this paper, unlike the cross-Gram matrix of two different sequences which is not always a diagnostic tool, we define the controlled-Gram matrix of a sequence as a practical implement to diagnose that a given sequence is a controlled Bessel, frame or Riesz basis. Also, we discuss the cases that the operator associated to controlled Gram matrix will be bounded, invertible, Hilbert–Schmidt or a trace-class operator. Similar to standard frames, we present an explicit structure for controlled Riesz bases and show that every [Formula: see text]-controlled Riesz basis [Formula: see text] is in the form [Formula: see text], where [Formula: see text] is a bijective operator on [Formula: see text]. Furthermore, we propose an equivalent accessible condition to the sequence [Formula: see text] being a [Formula: see text]-controlled Riesz basis.

New characterizations of g-frames and g-Riesz bases

2018 ◽  
Vol 16 (06) ◽  
pp. 1850053 ◽  
Author(s):  
Dongwei Li ◽  
Jinsong Leng ◽  
Tingzhu Huang
Keyword(s):  
Riesz Basis ◽  
Riesz Bases ◽  
Gram Matrix ◽  
Necessary Condition ◽  
Topological Properties ◽  
Bessel Sequence ◽  
Bessel Sequences ◽  
Sufficient And Necessary Condition

In this paper, we give some new characterizations of g-frames, g-Bessel sequences and g-Riesz bases from their topological properties. By using the Gram matrix associated with the g-Bessel sequence, we present a sufficient and necessary condition under which the sequence is a g-Bessel sequence (or g-Riesz basis). Finally, we consider the excess of a g-frame and obtain some new results.

scholarly journals GENERAL FRAMEWORK FOR CONSISTENT SAMPLING IN HILBERT SPACES

2005 ◽  
Vol 03 (04) ◽  
pp. 497-509 ◽  
Author(s):  
YONINA C. ELDAR ◽  
TOBIAS WERTHER
Keyword(s):  
Hilbert Spaces ◽  
General Framework ◽  
Inner Product ◽  
Riesz Bases ◽  
Oblique Projection ◽  
Dual Frames ◽  
Infinite Dimensional ◽  
Reconstruction Scheme ◽  
Linear Reconstruction ◽  
Oblique Dual

We introduce a general framework for consistent linear reconstruction in infinite-dimensional Hilbert spaces. We study stable reconstructions in terms of Riesz bases and frames, and generalize the notion of oblique dual frames to infinite-dimensional frames. As we show, the linear reconstruction scheme coincides with the so-called oblique projection, which turns into an ordinary orthogonal projection when adapting the inner product. The inner product of interest is, in general, not unique. We characterize the inner products and corresponding positive operators for which the new geometrical interpretation applies.

ON BOHR'S INEQUALITY

2002 ◽  
Vol 85 (2) ◽  
pp. 493-512 ◽  
Author(s):  
VERN I. PAULSEN ◽  
GELU POPESCU ◽  
DINESH SINGH
Keyword(s):  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Banach Algebras ◽  
Equivalent Norm ◽  
C Algebra ◽  
Theoretic Proof ◽  
Higher Dimensional ◽  
Bohr’S Inequality ◽  
Kernel Hilbert Spaces ◽  
Multiplier Algebras

Bohr's inequality says that if $f(z) = \sum^{\infty}_{n = 0} a_n z^n$ is a bounded analytic function on the closed unit disc, then $\sum^{\infty}_{n = 0} \lvert a_n\rvert r^n \leq \Vert f\Vert_{\infty}$ for $0 \leq r \leq 1/3$ and that $1/3$ is sharp. In this paper we give an operator-theoretic proof of Bohr's inequality that is based on von Neumann's inequality. Since our proof is operator-theoretic, our methods extend to several complex variables and to non-commutative situations.We obtain Bohr type inequalities for the algebras of bounded analytic functions and the multiplier algebras of reproducing kernel Hilbert spaces on various higher-dimensional domains, for the non-commutative disc algebra ${\mathcal A}_n$, and for the reduced (respectively full) group C*-algebra of the free group on $n$ generators.We also include an application to Banach algebras. We prove that every Banach algebra has an equivalent norm in which it satisfies a non-unital version of von Neumann's inequality.2000 Mathematical Subject Classification: 47A20, 47A56.

scholarly journals GENERAL FRAMEWORK FOR CONSISTENT SAMPLING IN HILBERT SPACES

2005 ◽  
Vol 03 (03) ◽  
pp. 347-359 ◽  
Author(s):  
YONINA C. ELDAR ◽  
TOBIAS WERTHER
Keyword(s):  
Hilbert Spaces ◽  
General Framework ◽  
Inner Product ◽  
Riesz Bases ◽  
Oblique Projection ◽  
Dual Frames ◽  
Infinite Dimensional ◽  
Reconstruction Scheme ◽  
Linear Reconstruction ◽  
Oblique Dual

We introduce a general framework for consistent linear reconstruction in infinite-dimensional Hilbert spaces. We study stable reconstructions in terms of Riesz bases and frames, and generalize the notion of oblique dual frames to infinte-dimensional frames. As we show, the linear reconstruction scheme coincides with the so-called oblique projection, which turns into an ordinary orthogonal projection when adapting the inner product. The inner product of interest is, in general, not unique. We characterize the inner products and the corresponding positive operators for which this geometrical interpretation applies.

scholarly journals Decompositions ofg-Frames and Duals and Pseudoduals ofg-Frames in Hilbert Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Xunxiang Guo
Keyword(s):  
Hilbert Spaces ◽  
Riesz Bases ◽  
Bounded Operators ◽  
Linear Combinations ◽  
Orthonormal Bases ◽  
Critical Components

Firstly, we study the representation ofg-frames in terms of linear combinations of simpler ones such asg-orthonormal bases,g-Riesz bases, and normalized tightg-frames. Then, we study the dual and pseudodual ofg-frames, which are critical components in reconstructions. In particular, we characterize the dualg-frames in a constructive way; that is, the formulae for dualg-frames are given. We also give someg-frame like representations for pseudodualg-frame pairs. The operator parameterizations ofg-frames and decompositions of bounded operators are the key tools to prove our main results.

scholarly journals Some New Characterizations andg-Minimality and Stability ofg-Bases in the Hilbert Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Xunxiang Guo
Keyword(s):  
Hilbert Spaces ◽  
Frame Theory ◽  
Schauder Basis ◽  
Simple Condition ◽  
Similar Role ◽  
Perturbation Result

The concept ofg-basis in the Hilbert spaces is introduced by Guo (2012) who generalizes the Schauder basis in the Hilbert spaces.g-basis plays the similar role ing-frame theory to that the Schauder basis plays in frame theory. In this paper, we establish some important properties ofg-bases in the Hilbert spaces. In particular, we obtain a simple condition under which some important properties established in Guo (2012) are still true. With these conditions, we also establish some new interesting properties ofg-bases which are related tog-minimality. Finally, we obtain a perturbation result aboutg-bases.

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