Operators In Spaces Research Articles

Spectrum of the Cesàro-Hardy operator in rearrangement invariant spaces

In this paper, we investigate the spectrum of the Cesàro-Hardy operator in rearrangement invariant spaces over a finite interval and a half line, thereby extending Boyd’s and Leibowitz’s results for Lp(1 < p ⩽ 8) spaces. In particular, when our rearrangement invariant space is the Lorentz space Lp,q , a full description of the spectrum and fine spectra is presented.

The Convergence of Some Spectral Characteristics on the Convergent Series

In this study, convergence properties of spectral, numerical and Crawford gap functions via convergences of Hilbert space operator series in difference and ratio cases are investigated. Obtained results have been applied to some classes continuous functions of the operators.

Generalized Euclidean operator radius

Abstract In this paper, we introduce the f-operator radius of Hilbert space operators as a generalization of the Euclidean operator radius and the q-operator radius. The properties of the newly defined radius are discussed, emphasizing how it extends some known results in the literature.

Construction of Intelligent Practice System of Visual Arts Based on VR Technology in Information Age

Abstract Taking visual art intelligence as an entry point, this paper describes the transformation brought by the information age to the expression of visual art. Based on VR technology to build a practice system of visual art intelligence, VR positioning technology is used to locate the three-dimensional coordinates of the operator in order to realize the independent positioning of multiple operators in a large space and avoid positioning interference. Through the eye movement multi-channel interaction algorithm based on head-eye synergy, the two-dimensional interaction task in flow field visualization is completed, and then gesture interaction is added to complete the three-dimensional interaction task. The constructed system was implemented in the teaching of art majors at College Z, and the data were analyzed. The results show that the total average scores of innovative theoretical ideas, interactive experience feeling, symbolic integration of creative content and infinite extension of the multidimensional spatial field are 82.42, 83.57, 80.84 and 80.28, respectively. The data show that teachers, students and parents recognize the intelligent system of visual arts and believe that it improves students’ innovative ability in art creation, enhances the interactive experience feeling of visual arts, and provides data support for the continuous research and exploration of contemporary visual arts.

Hierarchical neural architecture search with adaptive global–local feature learning for Magnetic Resonance Image reconstruction

Neural architecture search (NAS) has been introduced into the design of deep neural network architectures for Magnetic Resonance Imaging (MRI) reconstruction since NAS-based methods can acquire the complex network architecture automatically without professional designing experience and improve the model’s generalization ability. However, current NAS-based MRI reconstruction methods suffer from a lack of efficient operators in the search space, which leads to challenges in effectively recovering high-frequency details. This limitation is primarily due to the prevalent use of convolution operators in the current search space, which struggle to capture both global and local features of MR images simultaneously, resulting in insufficient information utilization. To address this issue, a generative adversarial network (GAN) based model is proposed to reconstruct the MR image from under-sampled K-space data. Firstly, parameterized global and local feature learning modules at multiple scales are added into the search space to improve the capability of recovering high-frequency details. Secondly, to mitigate the increased search time caused by the augmented search space, a hierarchical NAS is designed to learn the global–local feature learning modules that enable the reconstruction network to learn global and local information of MR images at different scales adaptively. Thirdly, to reduce the number of network parameters and computational complexity, the standard operations in global–local feature learning modules are replaced with lightweight operations. Finally, experiments on several publicly available brain MRI image datasets evaluate the performance of the proposed method. Compared to the state-of-the-art MRI reconstruction methods, the proposed method yields better reconstruction results in terms of peak signal-to-noise ratio and structural similarity at a lower computational cost. Additionally, our reconstruction results are validated through a brain tumor classification task, affirming the practicability of the proposed method. Our code is available at https://github.com/wwHwo/HNASMRI.

Some inequalities about Bourin's question in noncommutative L p -spaces

This paper studies a question of Bourin on noncommutative spaces. Some inequalities due to Hayajneh et al. [Norm inequalities for positive semidefinite matrices and a question of Bourin II. Int J Math. 2021;32:2150043] and Hayajneh et al. [Norm inequalities for positive semidefinite matrices and a question of Bourin III. Positivity. 2022;26:23] are extended to the case of operators in noncommutative spaces associated with a semifinite von Neumann algebra. We give a partial answer to a question of Bourin on noncommutative spaces. Some related submajorization inequalities are also given.

Classification of Ideals in Banach Spaces

Let an operator T belong to an operator ideal J, then for any operators A and B which can be composed with T asBTA then BTA \(\in\) J. Indeed, J contains the class of finite rank Banach Space operators. Now given L(X; Y ). Then J(X; Y ) \(\subseteq\) L(X; Y ) such that J(X; Y ) = {T : X \(\gets\) Y : T \(\subseteq\) }. Thus an operator ideal is a subclass J of L containing every identity operator acting on a one-dimensional Banach space such that: S + T \(\in\) J(X; Y ) where S; T \(\in\) J(X; Y ). If W;Z;X; Y \(\in\) K;A \(\in\) L(W;X);B \(\in\) L(Y;Z) then BTA \(\in\) J(W;Z) whenever T \(\in\) J(X; These properties compare very well with the algebraic notion of ideals in Banach Algebras within whose classes lie compact operators, weakly compact operators, finitely strictly regular operators, completely continuous operators, strictly singular operators among others. Thus, the aim of this paper is to characterize the various classes of ideals in Banach spaces. Special attention is given to the characteristics involving the ideal properties, the metric approximation properties, the hereditary properties in relation to the ideal extensions in the Hahn-Banach space, projection and embedments in the biduals of the Banach Space.

On Certain Spaces of Ideal Operators

We determine some important spaces of ideal operators and ideal characteristics. Special considerationis given to Frechet spaces, Spaces of finite rank operators and spaces of Hahn-Banach extension operators.The characteristics of ideals and related properties in these spaces as well as in some of their dual spacesare obtained.

Non-rough Norms and Dentability in Spaces of Operators

Non-rough Norms and Dentability in Spaces of Operators

Well‐posedness of degenerate fractional differential equations with finite delay in complex Banach spaces

AbstractWe study the well‐posedness of the degenerate fractional differential equations with finite delay: on Lebesgue–Bochner spaces and periodic Besov spaces , where and are closed linear operators in a complex Banach space satisfying , and are fixed, when , and the delay operator is a bounded linear operator from (resp. ) into . Using known operator‐valued Fourier multiplier theorems on and , we completely characterize the ‐well‐posedness and the ‐well‐posedness of above equations. We also give concrete examples that our abstract results may be applied.

Differentiable Ostrowski type tensorial norm inequality for continuous functions of selfadjoint operators in Hilbert spaces

In this paper several tensorial norm inequalities of Ostrowski type for continuous functions of selfadjoint operators in Hilbert spaces have been obtained. Multiple inequalities are obtained with variations due to the convexity properties of the mapping f.

Self-adaptive CQ-type algorithms for the split feasibility problem involving two bounded linear operators in Hilbert spaces

In this article, we consider and investigate a split convex feasibility problem involving two bounded linear operators in Hilbert spaces. We introduce a self-adaptive CQ-type algorithm by selecting the stepsize which is independent of the operator norms and establish a strong convergence result of the proposed algorithm under some mild control conditions. Moreover, we propose a self-adaptive relaxed CQ-type algo- rithm for solving the problem constrained by sub-level sets of convex functions. A numerical example and an application in compressed sensing are also given to illustrate the convergence behaviour of our proposed algorithms. Our results in this paper improve and generalize some existing results in the literature

Sixth-order finite difference schemes for nonlinear wave equations with variable coefficients in three dimensions

First, a nonlinear difference scheme is proposed to solve the three-dimensional (3D) nonlinear wave equation by combining the correction technique of truncation error remainder in time and a sixth-order finite difference operator in space, resulting in fourth-order accuracy in time and sixth-order accuracy in space. Then, the Richardson extrapolation method is applied to improve the temporal accuracy from the fourth-order to the sixth-order. To enhance computational efficiency, a linearized difference scheme is obtained by linear interpolation based on the nonlinear scheme. In addition, the stability of the linearized scheme is proved. Finally, the accuracy, stability and efficiency of the two proposed schemes are tested numerically.

Realization of Dirac quantization in loop quantum gravity

The system of gravity coupled to the non-rotational dust field is studied at both classical and quantum levels. The scalar constraint of the system can be written in the form of a true physical Hamiltonian with respect to the dust time. In the framework of loop quantum gravity, the scalar constraint is promoted to two versions of well-defined operators in a suitable Hilbert space of the coupled system, such that the physical Hamiltonian can become some symmetric operators. By the deparametrized form, a general expression of the solutions to either of the quantum scalar constraints is obtained, and the observables on the space of solutions can be constructed. Thus, the Dirac quantization procedure can be fully carried out in loop quantum gravity by this system.

Representations of frames via iterative actions of operators in tensor product spaces

Representations of frames via iterative actions of operators in tensor product spaces

Supported approach spaces

In this paper we work in the category of approach spaces with contractions [14], the objects of which are sets endowed with a numerical distance between sets and points. Approach spaces are to be considered a simultaneous generalization of both quasi-metric and topological spaces. Especially the fundamental notion of distance is reminiscent of the closure operator in a topological space and of the point-to-set distance in a quasi-metric space. The embedding of the category of topological spaces with continuous maps and of quasi-metric spaces with non-expansive maps is extremely nice. Every approach space has both a quasi-metric coreflection as well as a topological coreflection. Different approach spaces though can have the same topological as well as the same quasi-metric coreflection, in other words, in general these coreflections do not determine the approach space. In this paper we investigate approach spaces for which these coreflections, do determine the approach space. We will call such spaces supported. We prove that in the setting of compact approach spaces many examples of supported approach spaces can be found. Thus, compact spaces that are base-regular, which is a weakening of regularity, are always supported. An important feature of supported approach spaces is the behaviour of contractions. On a supported domain contractivity is characterized by the combination of continuity for the topological coreflection and non-expansiveness for the quasi-metric coreflection. This result implies that a supported approach space actually is the infimum of its quasi-metric and its topological coreflection. In the course of our study we also give several more examples of both supported and non-supported approach spaces.

$A$-numerical radius : New inequalities and characterization of equalities

We develop new lower bounds for the $A$-numerical radius of semi-Hilbertian space operators, and applying these bounds we obtain upper bounds for the $A$-numerical radius of the commutators of operators. The bounds obtained here improve on the existing ones. Further, we provide characterizations for the equality of the existing $A$-numerical radius inequalities of semi-Hilbertian space operators.

Bilinear forms, Schur multipliers, complete boundedness and duality

Grothendieck's inequalities for operators and bilinear forms imply some factorization results for complex $m \times n$ matrices. Based on the theory of operator spaces and completely bounded maps we present norm optimal versions of these results and two norm optimal factorization results related to the Schur product. We show that the spaces of bilinear forms and of Schur multipliers are conjugate duals to each other with respect to their completely bounded norms.

An efficient Julia framework for hierarchical equations of motion in open quantum systems

The hierarchical equations of motion (HEOM) approach can describe the reduced dynamics of a system simultaneously coupled to multiple bosonic and fermionic environments. The complexity of exactly describing the system-environment interaction with the HEOM method usually results in time-consuming calculations and a large memory cost. Here, we introduce an open-source software package called HierarchicalEOM.jl: a Julia framework integrating the HEOM approach. HierarchicalEOM.jl features a collection of methods to compute bosonic and fermionic spectra, stationary states, and the full dynamics in the extended space of all auxiliary density operators (ADOs). The required handling of the ADOs multi-indexes is achieved through a user-friendly interface. We exemplify the functionalities of the package by analyzing a single impurity Anderson model, and an ultra-strongly coupled charge-cavity system interacting with bosonic and fermionic reservoirs. HierarchicalEOM.jl achieves a significant speedup with respect to the corresponding method in the Quantum Toolbox in Python (QuTiP), upon which this package is founded.

Интегральное уравнение Вольтерра в банаховом пространстве

Despite the significant results achieved in the study of operator equations (including Volterra equations) in normalized Banach spaces, fundamental research in this field of mathematics attracts the attention of a huge number of mathematicians around the world. The solutions of the Volterra equation describe many important processes in various fields of science and technology. Studies of various inverse problems, experimental or experimental data processing problems related to the study of spherical or axisymmetric plasma formations, numerous mathematical models of the existence of various biological systems lead to the consideration and solution of this type of integral equations. A great contribution to the development of this theory was made by N.A. Magnitsky, L.I. Panov, A.N. Tikhonov, M.M. Lavrentiev and others. Fundamental results were obtained in the study of multiple operator equations with singularities in various functional spaces. Solutions depending on many parameters were constructed for the above equations. Currently, such problems are considered in spaces of arbitrary dimension and with coefficients having a derivative of finite order. In this paper, a finite set of solutions in a certain functional space is constructed for an integral equation of the first kind. The kernel of the integral operator has a finite order and is sufficiently differentiable near zero. The integral equation under consideration is reduced to an integro-differential equation representing two terms. For the first term, it is possible to solve the corresponding inhomogeneous equation and obtain a set of solutions in some functional normalized space. For the second term, we obtain an equation with an operator whose norm in some operator space is arbitrarily small near zero. Such splitting of the integral operator makes it possible to construct a partial and general solution of the corresponding integro-differential equation in the form of convergent equations. Applying modern methods of functional analysis, it is possible, by studying two separate equations, to construct a multiparametric family of solutions with values in some Banach space with weight for the original equation under consideration.