scholarly journals Invertibility of multipliers in Hilbert C*-modules

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 6073-6085
Author(s):  
Azandaryania Mirzaee
Keyword(s):  
Sufficient Conditions ◽  
Riesz Bases ◽  
Small Perturbations ◽  
Bessel Sequences ◽  
Bessel Multipliers

In this paper, we present some sufficient conditions under which Bessel multipliers in Hilbert C*-modules with semi-normalized symbols are invertible and we calculate the inverses. Especially we consider the invertibility of Bessel multipliers when the elements of their symbols are positive and when their Bessel sequences are equivalent, duals, modular Riesz bases or stable under small perturbations. We show that in these cases the inverse of a Bessel multiplier can be represented as a Bessel multiplier.

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.

Generalized frames for operators in Hilbert spaces

2014 ◽  
Vol 17 (02) ◽  
pp. 1450013 ◽  
Author(s):  
Mohammad Sadegh Asgari ◽  
Hamidreza Rahimi
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Separable Hilbert Space ◽  
Closed Subspace ◽  
Bounded Operator ◽  
Riesz Bases ◽  
Small Perturbations ◽  
Riesz Decomposition ◽  
Analysis And Synthesis ◽  
General Range

In this paper we present a family of analysis and synthesis systems of operators with frame-like properties for the range of a bounded operator on a separable Hilbert space. This family of operators is called a Θ–g-frame, where Θ is a bounded operator on a Hilbert space. Θ–g-frames are a generalization of g-frames, which allows to reconstruct elements from the range of Θ. In general, range of Θ is not a closed subspace. We also construct new Θ–g-frames by considering Θ–g-frames for its components. We further study Riesz decompositions for Hilbert spaces, which are a generalization of the notion of Riesz bases. We define the coefficient operators of a Riesz decomposition and we will show that these coefficient operators are continuous projections. We obtain some results about stability of Riesz decompositions under small perturbations.

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.

HILBERT–SCHMIDT OPERATORS AND FRAMES — CLASSIFICATION, BEST APPROXIMATION BY MULTIPLIERS AND ALGORITHMS

2008 ◽  
Vol 06 (02) ◽  
pp. 315-330 ◽  
Author(s):  
PETER BALAZS
Keyword(s):  
Hilbert Spaces ◽  
Best Approximation ◽  
Inner Product ◽  
Riesz Bases ◽  
Wavelet Frames ◽  
Arbitrary Matrix ◽  
The Best Approximation ◽  
Rank One ◽  
Arbitrary Operator ◽  
Bessel Sequences

In this paper we deal with the theory of Hilbert–Schmidt operators, when the usual choice of orthonormal basis, on the associated Hilbert spaces, is replaced by frames. We More precisely, we provide a necessary and sufficient condition for an operator to be Hilbert–Schmidt, based on its action on the elements of a frame (i.e. an operator T is [Formula: see text] if and only if the sum of the squared norms of T applied on the elements of the frame is finite). Also, we construct Bessel sequences, frames and Riesz bases of [Formula: see text] operators using tensor products of the same sequences in the associated Hilbert spaces. We state how the [Formula: see text] inner product of an arbitrary operator and a rank one operator can be calculated in an efficient way; and we use this result to provide a numerically efficient algorithm to find the best approximation, in the Hilbert–Schmidt sense, of an arbitrary matrix, by a so-called frame multiplier (i.e. an operator which act diagonally on the frame analysis coefficients). Finally, we give some simple examples using Gabor and wavelet frames, introducing in this way wavelet multipliers.

scholarly journals Time-periodic measures, random periodic orbits, and the linear response for dissipative non-autonomous stochastic differential equations

2021 ◽  
Vol 8 (3) ◽  
Author(s):  
Michał Branicki ◽  
Kenneth Uda
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Periodic Orbits ◽  
Linear Response ◽  
Sufficient Conditions ◽  
Probability Measures ◽  
Climate Modelling ◽  
Small Perturbations ◽  
Asymptotic Probability ◽  
Time Periodic

AbstractWe consider a class of dissipative stochastic differential equations (SDE’s) with time-periodic coefficients in finite dimension, and the response of time-asymptotic probability measures induced by such SDE’s to sufficiently regular, small perturbations of the underlying dynamics. Understanding such a response provides a systematic way to study changes of statistical observables in response to perturbations, and it is often very useful for sensitivity analysis, uncertainty quantification, and improving probabilistic predictions of nonlinear dynamical systems, especially in high dimensions. Here, we are concerned with the linear response to small perturbations in the case when the time-asymptotic probability measures are time-periodic. First, we establish sufficient conditions for the existence of stable random time-periodic orbits generated by the underlying SDE. Ergodicity of time-periodic probability measures supported on these random periodic orbits is subsequently discussed. Then, we derive the so-called fluctuation–dissipation relations which allow to describe the linear response of statistical observables to small perturbations away from the time-periodic ergodic regime in a manner which only exploits the unperturbed dynamics. The results are formulated in an abstract setting, but they apply to problems ranging from aspects of climate modelling, to molecular dynamics, to the study of approximation capacity of neural networks and robustness of their estimates.

Signal Representations Via SIP p-frames and SIP Bessel Multipliers in Separable Banach Spaces

2021 ◽  
pp. 2150005
Author(s):  
Xianwei Zheng ◽  
Cuiming Zou ◽  
Shouzhi Yang
Keyword(s):  
Banach Spaces ◽  
Frame Theory ◽  
Inner Product ◽  
Riesz Bases ◽  
Digital Signals ◽  
Bessel Sequence ◽  
Fixed Sequence ◽  
Signal Representations ◽  
Special Cases ◽  
Bessel Multipliers

Digital signals are often modeled as functions in Banach spaces, such as the ubiquitous [Formula: see text] spaces. The frame theory in Banach spaces induces flexible representations of signals due to the robustness and redundancy of frames. Nevertheless, the lack of inner product in general Banach spaces limits the direct representations of signals in Banach spaces under a given basis or frame. In this paper, we introduce the concept of semi-inner product (SIP) [Formula: see text]-Bessel multipliers to extend the flexibility of signal representations in separable Banach spaces, where [Formula: see text]. These multipliers are defined as composition of analysis operator of an SIP-I Bessel sequence, a multiplication with a fixed sequence and synthesis operator of an SIP-II Bessel sequence. The basic properties of the SIP [Formula: see text]-Bessel multipliers are investigated. Moreover, as special cases, characterizations of [Formula: see text]-Riesz bases related to signal representations are given, and the multipliers for [Formula: see text]-Riesz bases are discussed. We show that SIP [Formula: see text]-Bessel multipliers for [Formula: see text]-Riesz bases are invertible. Finally, the continuity of SIP [Formula: see text]-Bessel multipliers with respect to their parameters is investigated. The results theoretically show that the SIP [Formula: see text]-Bessel multipliers offer a larger range of freedom than frames on signal representations in Banach spaces.

Bessel Sequences and Its F-Scalability

2015 ◽  
Vol 7 (4) ◽  
pp. 441-453 ◽  
Author(s):  
Lei Liu ◽  
Xianwei Zheng ◽  
Jingwen Yan ◽  
Xiaodong Niu
Keyword(s):  
Quantum Mechanics ◽  
Wavelet Analysis ◽  
Frame Theory ◽  
Riesz Bases ◽  
Gabor Analysis ◽  
Perturbation Result ◽  
Extension Principles ◽  
Bessel Sequences

AbstractFrame theory, which contains wavelet analysis and Gabor analysis, has become a powerful tool for many applications of mathematics, engineering and quantum mechanics. The study of extension principles of Bessel sequences to frames is important in frame theory. This paper studies transformations on Bessel sequences to generate frames and Riesz bases in terms of operators and scalability. Some characterizations of operators that mapping Bessel sequences to frames and Riesz bases are given. We introduce the definitions of F-scalable and P-scalable Bessel sequences. F-scalability and P-scalability of Bessel sequences are discussed in this paper, then characterizations of scalings of F-scalable or P-scalable Bessel sequences are established. Finally, a perturbation result on F-scalable Bessel sequences is derived.

scholarly journals An algorithm for constructing multidimensional biorthogonal periodic multiwavelets

2000 ◽  
Vol 43 (3) ◽  
pp. 633-649 ◽  
Author(s):  
Say Song Goh ◽  
K. M. Teo
Keyword(s):  
Entire Function ◽  
Function Space ◽  
Sufficient Conditions ◽  
Riesz Bases ◽  
Extension Problem ◽  
The Matrix ◽  
Matrix Extension

AbstractThis paper deals with the problem of constructing multidimensional biorthogonal periodic multiwavelets from a given pair of biorthogonal periodic multiresolutions. Biorthogonal polyphase splines introduced to reduce the problem to a matrix extension problem, and an algorithm for solving the matrix extension problem is derived. Sufficient conditions for collections of periodic multiwavelets to form a pair of biorthogonal Riesz bases of the entire function space are also obtained.

Zur Stabilität eines Plasmas

1957 ◽  
Vol 12 (10) ◽  
pp. 833-841 ◽  
Author(s):  
K. Hain ◽  
R. Lust ◽  
A. Schlüter
Keyword(s):  
Magnetic Field ◽  
Differential Equation ◽  
Thermal Conductivity ◽  
Sufficient Conditions ◽  
Plasma Cylinder ◽  
Small Perturbations ◽  
Second Order In Time ◽  
The Stability ◽  
Conditions Of Stability ◽  
The Magnetic Field

Die Stabilität von hydrodynamischen Gleichgewichtskonfigurationen wird mit Hilfe der Methode der kleinen Störungen untersucht. Es wird gezeigt, daß das Stabilitätsverhalten durch eine Differentialgleichung 2. Ordnung in der Zeit bestimmt ist, wenn man die Viskosität, den elektrischen Widerstand und die thermische Leitfähigkeit vernachlässigt. Da die Differentialgleichung selbstadjungiert ist, können einige allgemeine Theoreme abgeleitet werden, welche für alle Gleichgewichtskonfigurationen gelten. Man kann zeigen, daß der zeitliche Anstieg von Störungen unter gewissen Bedingungen beschränkt ist. Weiterhin können einige hinreichende Bedingungen für die Stabilität angegeben werden. Für den Spezialfall, daß innerhalb eines Plasmazylinders das Magnetfeld verschwindet, werden die Differentialgleichungen explizit gelöst und Bedingungen für die Stabilität abgeleitet. Schließlich wird auch gezeigt, daß die Differentialgleichung auch selbstadjungiert ist, wenn der Druck nicht isotrop ist.It is shown that the stability of hydromagnetic equilibrium as studied by the method of small perturbations is controlled by one differential equation of second order in time, if one neglects viscosity, electrical resistivity and thermal conductivity. Since the differential equation is self-adjoint some general theorems can be derived which hold for all configurations of hydromagnetic equilibrium. It is possible to show that the rates of growing are limited under certain conditions. Also some sufficient conditions of stability can be given. For a plasma cylinder, inside of which the magnetic field vanishes, the differential equations are solved explicitly and conditions for stability are given. Finally it is shown that the differential equation is also self-adjoint if the pressure is not isotropic.

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