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.

Homeomorphic Images of Orthogonal Bases

2016 ◽  
Vol 31 ◽  
pp. 485-491
Author(s):  
M. Kebryaee ◽  
M. Radjabalipour
Keyword(s):  
Hilbert Space ◽  
Linear Algebra ◽  
Hilbert Spaces ◽  
Finite Subset ◽  
Sufficient Conditions ◽  
Orthogonal Basis ◽  
Riesz Bases ◽  
Orthonormal Bases ◽  
Homeomorphic Image ◽  
Necessary And Sufficient

Necessary and sufficient conditions are obtained for a sequence $\{x_j:~j\in \mathbb J\}$ in a Hilbert space to be, up to the elimination of a finite subset of $\mathbb J$, the linear homeomorphic image of an orthogonal basis of some Hilbert space $K$. This extends a similar result for orthonormal bases due to Holub [J.R. Holub. Pre-frame operators, Besselian frames, and near-Riesz bases in Hilbert spaces. \textit{Proc. Amer. Math. Soc.}, 122(3):779--785, 1994]. The proofs given here are based on simple linear algebra techniques.

scholarly journals Hermite Functions and Fourier Series

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 853
Author(s):  
Enrico Celeghini ◽  
Manuel Gadella ◽  
Mariano del Olmo
Keyword(s):  
Fourier Transform ◽  
Hermite Functions ◽  
Quantum Oscillator ◽  
Linear Combinations ◽  
Ladder Operators ◽  
Orthonormal Bases ◽  
Integrable Functions ◽  
The Fourier Transform ◽  
The One ◽  
Square Integrable Functions

Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle (L2(C)) and in l2(Z), which are related to each other by means of the Fourier transform and the discrete Fourier transform. These relations are unitary. The construction of orthonormal bases requires the use of the Gramm–Schmidt method. On both spaces, we have provided ladder operators with the same properties as the ladder operators for the one-dimensional quantum oscillator. These operators are linear combinations of some multiplication- and differentiation-like operators that, when applied to periodic functions, preserve periodicity. Finally, we have constructed riggings for both L2(C) and l2(Z), so that all the mentioned operators are continuous.

Infinitely entangled states

2003 ◽  
Vol 3 (4) ◽  
pp. 281-306
Author(s):  
M. Keyl ◽  
D. Schlingemann ◽  
R.F. Werner
Keyword(s):  
Hilbert Spaces ◽  
Degrees Of Freedom ◽  
Entangled States ◽  
Entangled State ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Density Operators ◽  
Fundamental Paper ◽  
Extended Notion

For states in infinite dimensional Hilbert spaces entanglement quantities like the entanglement of distillation can become infinite. This leads naturally to the question, whether one system in such an infinitely entangled state can serve as a resource for tasks like the teleportation of arbitrarily many qubits. We show that appropriate states cannot be obtained by density operators in an infinite dimensional Hilbert space. However, using techniques for the description of infinitely many degrees of freedom from field theory and statistical mechanics, such states can nevertheless be constructed rigorously. We explore two related possibilities, namely an extended notion of algebras of observables, and the use of singular states on the algebra of bounded operators. As applications we construct the essentially unique infinite analogue of maximally entangled states, and the singular state used heuristically in the fundamental paper of Einstein, Rosen and Podolsky.

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.

Three Test Problems for Quasisimilarity

1987 ◽  
Vol 39 (4) ◽  
pp. 880-892 ◽  
Author(s):  
Hari Bercovici
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Structure Theory ◽  
Test Problems ◽  
Unitary Equivalence ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Densely Defined

Kaplansky proposed in [7] three problems with which to test the adequacy of a proposed structure theory of infinite abelian groups. These problems can be rephrased as test problems for a structure theory of operators on Hilbert space. Thus, R. Kadison and I. Singer answered in [6] these test problems for the unitary equivalence of operators. We propose here a study of these problems for quasisimilarity of operators on Hilbert space. We recall first that two (bounded, linear) operators T and T′ acting on the Hilbert spaces and , are said to be quasisimilar if there exist bounded operators and with densely defined inverses, satisfying the relations T′X = XT and TY = YT′. The fact that T and T′ are quasisimilar is indicated by T ∼ T′. The problems mentioned above can now be formulated as follows.

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.

(F,G)-operator frames for ℒ(ℋ,𝒦)

2020 ◽  
Vol 18 (05) ◽  
pp. 2050031
Author(s):  
J. Sedghi Moghaddam ◽  
A. Najati ◽  
F. Ghobadzadeh
Keyword(s):  
Linear Operator ◽  
Hilbert Spaces ◽  
Bounded Linear Operator ◽  
Bounded Operators ◽  
Closed Range ◽  
Bounded Linear ◽  
Atomic Systems ◽  
Pseudo Inverse

The concept of [Formula: see text]-frames was recently introduced by Găvruta7 in Hilbert spaces to study atomic systems with respect to a bounded linear operator. Let [Formula: see text] be a unital [Formula: see text]-algebra, [Formula: see text] be finitely or countably generated Hilbert [Formula: see text]-modules, and [Formula: see text] be adjointable operators from [Formula: see text] to [Formula: see text]. In this paper, we study a class of [Formula: see text]-bounded operators and [Formula: see text]-operator frames for [Formula: see text]. We also prove that the pseudo-inverse of [Formula: see text] exists if and only if [Formula: see text] has closed range. We extend some known results about the pseudo-inverses acting on Hilbert spaces in the context of Hilbert [Formula: see text]-modules. Further, we also present some perturbation results for [Formula: see text]-operator frames in [Formula: see text].

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