Operator Norm Research Articles

L2 bounds for a maximal directional Hilbert transform

Given any finite direction set $\Omega$ of cardinality $N$ in Euclidean space, we consider the maximal directional Hilbert transform $H_{\Omega}$ associated to this direction set. Our main result provides an essentially sharp uniform bound, depending only on $N$, for the $L^2$ operator norm of $H_{\Omega}$ in dimensions 3 and higher. The main ingredients of the proof consist of polynomial partitioning tools from incidence geometry and an almost-orthogonality principle for $H_{\Omega}$. The latter principle can also be used to analyze special direction sets $\Omega$, and derive sharp $L^2$ estimates for the corresponding operator $H_{\Omega}$ that are typically stronger than the uniform $L^2$ bound mentioned above. A number of such examples are discussed.

Operators of the alfa-Bloch space on the open unit ball of a JB*-triple

"Let $\B_X$ be a bounded symmetric domain realized as the open unit ball of a JB*-triple $X$ which may be infinite dimensional. In this paper, we characterize the bounded weighted composition operators from the Hardy space $H^{\infty}(\mathbb{B}_X)$ into the $\alpha $-Bloch space $\mathcal{B}^\alpha (\B_X)$ on $\mathbb{B}_X$. Later, we show the multiplication operator from $H^{\infty}(\mathbb{B}_X)$ into $\mathcal{B}^\alpha (\B_X)$ is bounded. Also, we give the operator norm of the bounded multiplication operator."

Testing for implicit bias: Values, psychometrics, and science communication.

Our understanding of implicit bias and how to measure it has yet to be settled. Various debates between cognitive scientists are unresolved. Moreover, the public's understanding of implicit bias tests continues to lag behind cognitive scientists'. These discrepancies pose potential problems. After all, a great deal of implicit bias research has been publicly funded. Further, implicit bias tests continue to feature in discourse about public- and private-sector policies surrounding discrimination, inequality, and even the purpose of science. We aim to do our part by reconstructing some of the recent arguments in ordinary language and then revealing some of the operative norms or values that are often hidden beneath the surface of these arguments. This may help the public learn more about the science of implicit bias. It may also help both laypeople and scientists reflect on the values, interests, and stakeholders involved in establishing, justifying, and communicating scientific research. This article is categorized under: Cognitive Biology > Social Development.

Uniform convergence of Nyström discretization on Hölder spaces

We establish that Nyström discretizations of linear Fredholm integral operators on Hölder spaces converge in the operator norm while preserving the consistency order of the quadrature or cubature rule. This allows to employ tools from classical perturbation theory, rather than collective compactness, when studying numerical approximations of integral operators, as well as applications in for instance the field of nonautonomous dynamical systems.

Cut norm discontinuity of triangular truncation of graphons

The space of Lp graphons, symmetric measurable functions w:[0,1]2→R with finite p-norm, features heavily in the study of sparse graph limit theory. We show that the triangular cut operator Mχ—the operator that sets all values of a graphon below the main diagonal of the unit square to 0—acting on this space is not continuous with respect to the cut norm. This is achieved by showing that as n→∞, the norm of the triangular truncation operator Tn on symmetric matrices equipped with the cut norm grows to infinity as well. Due to the density of symmetric matrices in the space of Lp graphons, the norm growth of Tn generalizes to the unboundedness of Mχ. We also show that the norm of Tn grows to infinity on symmetric matrices equipped with the operator norm.

Positive Invertibility of Matrices and Exponential Stability of Linear Stochastic Systems with Delay

The work addresses the exponential moment stability of solutions of large systems of linear differential Itô equations with variable delays by means of a modified regularization method, which can be viewed as an alternative to the technique based on Lyapunov or Lyapunov-like functionals. The regularization method utilizes the parallelism between Lyapunov stability and input-to-state stability, which is well established in the deterministic case, but less known for stochastic differential equations. In its practical implementation, the method is based on seeking an auxiliary equation, which is used to regularize the equation to be studied. In the final step, estimation of the norm of an integral operator or verification of the property of positivity of solutions is performed. In the latter case, one applies the theory of positive invertible matrices. This report contains a systematic presentation of how the regularization method can be applied to stability analysis of linear stochastic delay equations with random coefficients and random initial conditions. Several stability results in terms of positive invertibility of certain matrices constructed for general stochastic systems with delay are obtained. A number of verifiable sufficient conditions for the exponential moment stability of solutions in terms of the coefficients for specific classes of Itô equations are offered as well.

An accelerate algorithm for the split equality common fixed-point problem of directed operators

In this paper, we introduce an inertial accelerated algorithm for solving the split equality common fixed-point problem of directed operators in real Hilbert space. Our algorithm includes the simultaneous iterative algorithm as special case which has been proposed by Moudafi and Al-Shemas for solving the split equality common fixed-point problem. We establish a weak convergence theorem for the proposed iterative algorithm, which combines the primal-dual method and the inertial technique. In our algorithm, the step sizes are chosen self-adaptively so that the implementation of the algorithm does not need any prior information about bounded linear operator norms. The efficiency of the proposed algorithm is illustrated by some numerical experiments.

Experts' consensus on perioperative management of tooth extractions in patients receiving oral antithrombotic treatment.

Thromboembolic diseases, which comprise venous thromboembolic diseases and arterial thromboembolic diseases, have become the number one cause of death worldwide. To prevent or treat thrombosis, patients with thromboembolic diseases need to take antithrombotic drugs, which would increase the risk of bleeding during and after surgery. Tooth extraction is the most common operation in oral and maxillofacial surgery clinics. Although patients given oral antithrombotic drugs do not need to undergo drug withdrawal, the perioperative management of such patients remains confusing to most clinicians. Moreover, the potential risk factors for bleeding warrant further study. To improve the clinicians' knowledge of perioperative management for patients subjected to tooth extractions with oral antithrombotic drugs, experts have drafted this consensus focusing on preoperative bleeding risk assessment, intraoperative operating norms, and postoperative care to summarize the points needing attention.

On the convolution operator in Morrey spaces

This paper is devoted to the study of upper bounds for the norm of the convolution operator in Morrey spaces. The spaces Mp,qα, which cover the classical Morrey spaces, are introduced. Moreover, their embedding properties are investigated, and their interpolation properties are described. Young-O'Neil type inequalities in Morrey spaces are proved. New results on the boundedness of Riesz's potential in Morrey spaces are established.

Norm and numerical radius of single operators through tools and techniques from multivariable operator theory

We employ tools and techniques from multivariable operator theory to obtain new proofs and extensions of well known inequalities regarding the norm and the numerical radius of elementary operators defined on the C⁎–algebra of all bounded operators on Hilbert space, or on the ⁎–ideal of Hilbert-Schmidt operators. In the process, we provide new insights on the study of Heinz-type inequalities related to the arithmetic-geometric mean inequality, and generalize results of several authors, including R. Bhatia, G. Corach, C. Davis, F. Kittaneh, and M.S. Moslehian. To estimate the norm, our approach exploits, in particular, the Spectral Mapping Theorem for the Taylor spectrum, and Ky Fan's Dominance Theorem. For the numerical radius, we use S. Hildebrandt's description of the numerical range of an operator in terms of the norm of its translates.

Polyharmonic hypersurfaces into space forms

In this paper we shall consider polyharmonic hypersurfaces of order r (briefly, r-harmonic hypersurfaces), where r ≥ 3 is an integer, into a space form Nm+1 (c) of curvature c. For this class of hypersurfaces we shall prove that, if c ≤ 0, then any r-harmonic hypersurface must be minimal provided that the mean curvature function and the squared norm of the shape operator are constant. When the ambient space is $${\mathbb{S}^{m + 1}}$$ , we shall obtain the geometric condition which characterizes the r-harmonic hypersurfaces with constant mean curvature and constant squared norm of the shape operator, and we shall establish the bounds for these two constants. In particular, we shall prove the existence of several new examples of proper r-harmonic isoparametric hypersurfaces in $${\mathbb{S}^{m + 1}}$$ for suitable values of m and r. Finally, we shall show that all these r-harmonic hypersurfaces are also ES-r-harmonic, i.e., critical points of the Eells-Sampson r-energy functional.

G-Fuzzy Operator Norm

In our previous paper, it is shown that topology of [Formula: see text]-fuzzy normed linear space is generated by two types of open balls: one is elliptic and the other is circular. In the theoretical aspect of functional analysis, will this type of exception happen or not? To address this problem in this paper, firstly, [Formula: see text]-fuzzy bounded linear operators as well as [Formula: see text]-fuzzy bounded linear functionals are defined which are the key elements of functional analysis. Then, operator [Formula: see text]-fuzzy norms are introduced for both the cases using the idea of quasi-[Formula: see text]-norm family. The definition of operator [Formula: see text]-fuzzy norm is quite different from the existing operator fuzzy norm. Completeness of operator [Formula: see text]-fuzzy norm is investigated. Lastly, Hahn-Banach theorem in [Formula: see text]-fuzzy setting is studied using all the above concepts.

A COMPARATIVE ANALYSIS OF THE IMPACT OF THE COVID-19 PANDEMIC ON RURAL UNIVERSITY STUDENTS IN SOUTH AFRICA AND UGANDA

The outbreak of COVID-19 significantly caused shifts in higher education’s environment and operational norms. This has propelled new educational strategies, models and frameworks. Globally, COVID-19 wreaked havoc on institutions of higher learning, such as universities. This was a qualitative comparative and phenomenological study aimed at exploring the impact and implications of COVID-19 on rural university students in South Africa and Uganda. The services offered to rural communities have a bearing on educational institutions in those environments. The study further suggested possible contingency plans to counter the negative impacts caused by such pandemics on rural students in universities. The findings, through a thematic analysis, revealed that rural university students in the two countries were affected by COVID-19 instructional, psychological and social ways. In conclusion, rural university students have peculiar challenges due to their unique but disadvantaged settings. The study recommends that online teaching and learning be prioritized to benefit rural students, preceded by adequate training in Ed Tech skills for both lecturers and students. There is need for guidance and counselling psycho-social interventions for rural university students to address the related challenges.

Matrix dilation and Hausdorff operators on modulation spaces

In this paper, we establish the asymptotic estimates for the norms of the matrix dilation operators on modulation spaces. As an application, we study the boundedness on modulation spaces of Hausdorff operators. The definition of Hausdorff operators is also revisited for fitting our study.

Norm of the Hilbert matrix operator on the Korenblum space

In this paper, the norm representation of the Hilbert matrix operator H on the Korenblum space Hα∞ is given for 0<α<1. In particular, we find an exact value α0 in (0,1) such that the norm of H is equal to πsin⁡(πα) for 0<α≤α0, and the norm is greater than πsin⁡(πα) for α0<α<1.

When a system of real quadratic equations has a solution

We provide a sufficient condition for solvability of a system of real quadratic equations pi(x)=yi, i=1,…,m, where pi:Rn⟶R are quadratic forms. By solving a positive semidefinite program, one can reduce it to another system of the type qi(x)=αi, i=1,…,m, where qi:Rn⟶R are quadratic forms and αi=traceqi. We prove that the latter system has solution x∈Rn if for some (equivalently, for any) orthonormal basis A1,…,Am in the space spanned by the matrices of the forms qi, the operator norm of A12+…+Am2 does not exceed η/m for some absolute constant η>0. The condition can be checked in polynomial time and is satisfied, for example, for random qi provided m≤γn for an absolute constant γ>0. We prove a similar sufficient condition for a system of homogeneous quadratic equations to have a non-trivial solution. While the condition we obtain is of an algebraic nature, the proof relies on analytic tools including Fourier analysis and measure concentration.

The essential norm of multiplication operators on L_p(mu )

We show that the formula for the essential norm of a multiplication operator on L_p, for 1<p<infty , also holds for p=1. We also provide a proof for the formula which works simultaneously for all pin [1,infty ).

Multicriteria Optimization of Induced Norms of Linear Operators: Primal and Dual Control and Filtering Problems

Multicriteria Optimization of Induced Norms of Linear Operators: Primal and Dual Control and Filtering Problems

Conditions for the existence of smooth solutions for a class of fourth order operator-differential equations

. In the paper, we consider a fourth-order operator-differential equation on the entire axis, the main part of which has a multiple characteristic, and we introduce the concept of its ”smoothly” regular solvability. We find exact values of the norms of intermediate derivatives operators in a Sobolev-type space and indicate their connection with the conditions for the solvability of the equation under study. Note that the conditions found for ”smoothly” regular solvability are sufficient and are imposed only on the operator coefficients of the operator-differential equation under consideration

On Translation of Shaanxi Dialect-featured Works Based on Gideon Toury’s Theory of Translation Norms: A Case Study of Life and Its English Translation

This research analyzes Chloe Estep's translated version Life（《人生》）in a descriptive and explanatory way based on Gideon Toury's Translation Norms. By investigating the reason why Estep chose to translate this book under the influence of preliminary norm, the orientation of her translation governed by initial norm and specific translation strategies guided by operational norm, the paper intends to prove Translation Norms to some degree influence translating process and more importantly analyze Estep's strategy and skills on translating Shaanxi dialect-featured words and sentences. By investigating the forming mechanism behind this translated work, the paper hopes to make some contribution to translating Shaanxi dialect-featured literary works and descriptively analyzing the translated version of Chinese literature.