Commutators Of Operators Research Articles

Rings of almost everywhere defined functions

The following representation theorem is proven: A partially ordered commutative ring R is a subring of a ring of almost everywhere defined continuous real-valued functions on a compact Hausdorff space X if and only if R is archimedean and localizable. Here we assume that the positive cone of R is closed under multiplication and stable under multiplication with squares, but actually one of these assumptions implies the other. An almost everywhere defined function on X is one that is defined on a dense open subset of X. A partially ordered commutative ring R is archimedean if the underlying additive partially ordered abelian group is archimedean, and R is localizable essentially if its order is compatible with the construction of a localization with sufficiently large, positive denominators. As applications we discuss the σ-bounded case, lattice-ordered commutative rings (f-rings), partially ordered fields, and commutative operator algebras.

Pembentukan Representasi Adjoin pada Aljabar Lie

Algebra is a branch of mathematics that can provide understanding in solving problems. This research discusses one form of algebra, namely Lie algebra. Where Lie algebra 𝗀 is a vector space over the field 𝔽 equipped with a bilinear mapping equipped by a commutator operation commonly called the Lie bracket operation denoted [−, −] from 𝗀 × 𝗀 to 𝗀 if it satisfies the anti-symmetry axiom and satisfies Jacobi Identity. One model of Lie algebra is the set of all linear operators of a vector space 𝑉 denoted by 𝑔𝑙(𝑉). In addition, one of the discussions related to Lie algebra in this research is representation theory. Lie algebra uses representation theory with the aim of simplifying abstract algebraic problems into linear algebra by presenting each of its members in the form of linear mappings on vector spaces. There are various forms of representation theory in Lie algebra, one of which is adjoin representation. And this adjoin representation is formed from a derivation and Lie homomorphism. The purpose of this research is to find out how the formation of adjoin representation on Lie algebra. This research uses a qualitative method, where the method applies a way to collect data or library materials as a reference in the form of articles, journals, and even books related to Lie algebra.

Improvement on the screening of nonlinear commutator operations in selective coupled-cluster using Lagrangian.

We introduce a Lagrangian implementation of the full coupled-cluster reduction [Xu et al., Phys. Rev. Lett. 121, 113001 (2018)], that is, a selected coupled-cluster (CC) based on an arbitrary-order full CC expansion using direct commutator expansions. In this method, the screening for the products of cluster amplitudes plays a central role to reduce the computational cost for the nonlinear commutator operations, while the convergence of the total energy in the standard energy expression is not rapid with tightening the threshold. The new implementation using Lagrangian is robust, containing error only quadratic to those of amplitudes, allowing a much larger screening threshold. We demonstrate the performance of the new implementation by investigating the calculations of N2 and C6H6. The accuracy and applicability are also demonstrated for the potential energy curve of H2O in comparison with conventional quantum chemical methods.

State-dependent and state-independent uncertainty relations for skew information and standard deviation

In this work, we derive state-dependent uncertainty relations (uncertainty equalities) in which commutators of incompatible operators (not necessarily Hermitian) are explicitly present and state-independent uncertainty relations based on the Wigner-Yanase (-Dyson) skew information. We derive uncertainty equality based on standard deviation for incompatible operators with mixed states, a generalization of previous works in which only pure states were considered. We show that for pure states, the Wigner-Yanase skew information based state-independent uncertainty relations become standard deviation based state-independent uncertainty relations which turn out to be tighter uncertainty relations for some cases than the ones given in previous works, and we generalize the previous works for arbitrary operators. As the Wigner-Yanase skew information of a quantum channel can be considered as a measure of quantum coherence of a density operator with respect to that channel, we show that there exists a state-independent uncertainty relation for the coherence measures of the density operator with respect to a collection of different channels. We show that state-dependent and state-independent uncertainty relations based on a more general version of skew information called generalized skew information which includes the Wigner-Yanase (-Dyson) skew information and the Fisher information as special cases hold. In qubits, we derive tighter state-independent uncertainty inequalities for different form of generalized skew informations and standard deviations, and state-independent uncertainty equalities involving generalized skew informations and standard deviations of spin operators along three orthogonal directions. Finally, we provide a scheme to determine the Wigner-Yanase (-Dyson) skew information of an unknown observable which can be performed in experiment using the notion of weak values.

Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces

Abstract In this work, we present necessary and sufficient conditions for the boundedness of the commutators generated by multilinear fractional maximal operators on the products of Morrey spaces when the symbol belongs to Lipschitz spaces.

Comprehensive review on evolution, progression, design, and exploration of electric bicycle

Many countries are moving to electric vehicles for daily transportation due to the scarcity of fossil fuels and the pollution emissions from internal combustion engines. The most promising technique to manage environmental pollution and energy security is the employment of electric vehicles. The electric bicycle is gaining an abundance of attention due to the various advantages of electric vehicles. Recently, people started using electric bicycles on a regular basis, and it is a suitable option for shorter daily rides. The present study was conducted using the Scopus database, which included an analysis of all papers pertaining to the content of electric bicycles up until the year 2023.The literature review delves into the profound impact and numerous benefits of electric bicycles, highlighting their potential for revolutionizing transportation. This exploration is informed by advancements in design and construction, offering insights into optimal electric bicycle development. One promising avenue for sustainability involves integrating solar technology into these vehicles, reducing reliance on conventional energy sources, and addressing environmental concerns. Discussions encompassed diverse aspects, including operation principles, throttle variations, commutation systems, self-charging mechanisms, health advantages, and solar-powered electric bicycles. To address this, the goal is to create universally acceptable electric bicycles to broaden their appeal and dispel misconceptions. In order to provide scientists and researchers with beneficial information to further optimize electric bicycles, the research fields pertaining to electric bicycles are reviewed in accordance with previous studies.

On p-numerical radius inequalities for commutators of operators

In this work, we give some p-numerical radius inequalities for operators as well as for commutators of operators. As applications, when p = ∞, we improve some earlier numerical radius inequalities for operators. Also, we refine the triangle inequalities for the numerical radius and the operator norm.

Mixed radial-angular integrabilities for commutators of fractional Hardy operators

In this paper, we introduce the mixed radial-angular homogeneous CMO spaces CMOLradp2Langp1(Rn) and the mixed radial-angular Herz spaces K˙Lradp2Langp1α,q(Rn), respectively. Furthermore, the boundedness of commutators Hβ,b generated by the fractional Hardy operators Hβ with a function b in CMOLradp2Langp1(Rn) are established on K˙Lradp2Langp1α,q(Rn). Meanwhile, the corresponding results for Hβ,b when b∈CMOLradp2,λLangp1(Rn), the mixed radial-angular homogeneous λ-central bounded mean oscillation spaces, are also considered on BLradp2,λLangp1(Rn), the mixed radial-angular homogeneous λ-central Morrey spaces.

Sobolev-type theorem for commutators of Hardy operators in grand Herz spaces

Sobolev-type theorem for commutators of Hardy operators in grand Herz spaces

Weighted mixed endpoint estimates of Fefferman-Stein type for commutators of singular integral operators

We deal with mixed weak estimates of Fefferman-Stein type for higher order commutators of Calderón-Zygmund operators with BMO symbol. The results obtained are Fefferman-Stein inequalities that include the estimates proved in [7] for the case of singular integral operators, as well as the classical weak endpoint estimate for commutators given in [26]. We also consider commutators of operators involving less regular kernels satisfying an LΦ–Hörmander condition. Particularly, the obtained results contain some previous estimates proved in [7] and [17].

Compactness of commutators of Hardy operators on Heisenberg group

Compactness of commutators of Hardy operators on Heisenberg group

Commutators greater than a perturbation of the identity

Let a and b be elements of an ordered normed algebra A with unit e. Suppose that the element a is positive and that for some ε>0 there exists an element x∈A with ‖x‖≤ε such thatab−ba≥e+x. If the norm on A is monotone, then we show‖a‖⋅‖b‖≥12ln⁡1ε, which can be viewed as an order analog of Popa's quantitative result for commutators of operators on Hilbert spaces.We also give a relevant example of positive operators A and B on the Hilbert lattice ℓ2 such that their commutator AB−BA is greater than an arbitrarily small perturbation of the identity operator.

Population Fluctuation Mechanism of the Super‐Thermal Photon Statistic of Quantum LEDs with Collective Effects

AbstractThe analytical procedure is developed for the calculation of the quantum second‐order autocorrelation function of a small super‐radiant LED, where the field, polarisation, and population of the active medium cannot be eliminated adiabatically. The Langevin force, which describes the effect of population fluctuations on the LED polarisation and preserves the operator commutation relations, is found. It is demonstrated that the super‐thermal photon statistics with of a small quantum LED with large photon number fluctuations, emitter‐field coupling, bad cavity, and operating in the limit , is the result of a combined effect of the spontaneous emission, collective effects and population fluctuations. Analytical expressions for and are derived.

Commutativity of Hankel and Toeplitz operators on the Hardy space of the n-torus

We consider Hankel and Toeplitz operators on H2(Tn), the Hardy space of the n-torus Tn. Given symbols φ and ψ in L∞(Tn) with suitable properties, we obtain necessary and sufficient conditions for the Hankel operator Hψ,n and the Toeplitz operator Tφ,n to commute. We then extend the study to the more general situation where no assumptions are imposed on φ, and provide new, non-trivial necessary conditions for the commutativity of Hψ,n and Tφ,n. We also show that certain well known commutativity results between Hankel and Toeplitz operators in the one-variable case do not extend to the multivariable setting.

On commutators of idempotents

Let T be an operator on a Banach space X that is similar to − T via an involution U. Then,U decomposes the Banach space X as X = X 1 ⊕ X 2 with respect to which decomposition we have U = ( I 1 0 0 − I 2 ) , where I i is the identity operator on the closed subspace X i (i = 1, 2). Furthermore, T has necessarily the form T = ( 0 ∗ ∗ 0 ) with respect to the same decomposition. In this note, we consider the question when T is a commutator of the idempotent P = ( I 1 0 0 0 ) and some idempotent Q on X. We also determine which scalar multiples of unilateral shifts on l p spaces ( 1 ≤ p < ∞ ) are commutators of idempotent operators.

Commutativity and Compactness of kth Order Slant Toeplitz Operators

Commutativity and Compactness of kth Order Slant Toeplitz Operators

Weighted estimates of commutators of singular operators in generalized Morrey spaces beyond Muckenhoupt range and applications

For a certain class of radial weights, we prove weighted norm estimates for commutators with BMO coefficients of singular operators in local generalized Morrey spaces. As a consequence of these estimates, we obtain norm inequalities for such commutators in the generalized Stummel-Morrey spaces. We also discuss a.e. well-posedness of singular operators and their commutators on weighted generalized Morrey spaces. The obtained estimates are applied to prove interior regularity for solutions of elliptic PDEs in the frameworks of the corresponding weighted Sobolev spaces based on the local generalized Morrey spaces or Stummel-Morrey spaces. To this end also conditions for the applicability of the representation formula, for the second-order derivatives of solutions to elliptic PDEs, are found for the case of such weighted spaces. In both results, for commutators and applications, we admit weights beyond the Muckenhoupt range.

Characterizations of the mixed central Campanato spaces via the commutator operators of Hardy type

The purpose of this paper is to establish some characterizations of mixed central Campanato space Cp→,λ(Rn), via the boundedness of the commutator operators of Hardy type. Unlike the case λ⩾0, there are some technical difficulties caused by λ<0 to be overcome. In addition, an extra assumption called as the mixed version of the reverse Hölder class be required in the proof of the converse characterization. Moreover, some further interesting conclusions for the Hardy type operators on mixed λ-central Morrey space Bp→,λ(Rn) are also derived.

On Commutators of Compact Operators: Generalizations and Limitations of Anderson’s Approach

On Commutators of Compact Operators: Generalizations and Limitations of Anderson’s Approach

Extrapolation of Compactness on Banach Function Spaces

We prove an extrapolation of compactness theorem for operators on Banach function spaces satisfying certain convexity and concavity conditions. In particular, we show that the boundedness of an operator T in the weighted Lebesgue scale and the compactness of T in the unweighted Lebesgue scale yields compactness of T on a very general class of Banach function spaces. As our main new tool, we prove various characterizations of the boundedness of the Hardy-Littlewood maximal operator on such spaces and their associate spaces, using a novel sparse self-improvement technique. We apply our main results to prove compactness of the commutators of singular integral operators and pointwise multiplication by functions of vanishing mean oscillation on, for example, weighted variable Lebesgue spaces.