Operator Norm Research Articles

Invariant Gibbs dynamics for the two-dimensional Zakharov-Yukawa system

We study the Gibbs dynamics for the Zakharov-Yukawa system on the two-dimensional torus T2, namely a Schrödinger-wave system with a Zakharov-type coupling (−Δ)γ. We first construct the Gibbs measure in the weakly nonlinear coupling case (0≤γ<1). Combined with the non-construction of the Gibbs measure in the strongly nonlinear coupling case (γ=1) by Oh, Tolomeo, and the author (2020), this exhibits a phase transition at γ=1. We also study the dynamical problem and prove almost sure global well-posedness of the Zakharov-Yukawa system and invariance of the Gibbs measure under the resulting dynamics for the range 0≤γ<13. In this dynamical part, the main step is to prove local well-posedness. Our argument is based on the first order expansion and the operator norm approach via the random matrix/tensor estimate from a recent work Deng, Nahmod, and Yue (2020). In the appendix, we briefly discuss the Hilbert-Schmidt norm approach and compare it with the operator norm approach.

AN APPLICATION OF MAXIMAL NUMERICAL RANGE ON NORM OF AN ELEMENTARY OPERATOR OF LENGTH TWO IN TENSOR PRODUCT

Many researchers in operator theory have attempted to determine the relationship between the norm of an elementary operator of length two and the norms of its coefficient operators. Various results have been obtained using varied approaches. In this paper, we attempt this problem by the use of the Stampfli’s maximal numerical range in a tensor product.

Bounds on the norms of maximal operators on Weyl sums

We obtain new estimates on the maximal operator applied to the Weyl sums, which are tight in some cases. We also consider the quadratic case (that is, Gauss sums) in more details. In wide ranges of parameters our estimates are optimal and match lower bounds. Our approach is based on a combination of ideas of Baker (2021) and Chen and Shparlinski (2020).

Elliptic polytopes and invariant norms of linear operators

Elliptic polytopes and invariant norms of linear operators

New inertial self-adaptive algorithms for the split common null-point problem: application to data classifications

In this paper, we propose two inertial algorithms with a new self-adaptive step size for approximating a solution of the split common null-point problem in the framework of Banach spaces. The step sizes are adaptively updated over each iteration by a simple process without the prior knowledge of the operator norm of the bounded linear operator. Under suitable conditions, we prove the weak-convergence results for the proposed algorithms in p-uniformly convex and uniformly smooth Banach spaces. Finally, we give several numerical results in both finite- and infinite-dimensional spaces to illustrate the efficiency and advantage of the proposed methods over some existing methods. Also, data classifications of heart diseases and diabetes mellitus are presented as the applications of our methods.

ON SPLIT GENERALIZED EQUILIBRIUM AND FIXED POINT PROBLEMS OF BREGMAN W-MAPPINGS WITH MULTIPLE OUTPUT SETS IN REFLEXIVE BANACH SPACES

In this paper, we introduce a Halpern iteration process for computing the common solution of split generalized equilibrium problem and fixed points of a countable family of Bregman W-mappings with multiple output sets in reflexive Banach spaces. We prove a strong convergence result for approximating the solutions of the aforementioned problems under some mild conditions. It is worth mentioning that the iterative algorithm employ in this article is designed in such a way that it does not require the prior knowledge of operator norm. We also provide some numerical examples to illustrate the performance of our proposed iterative method. The result discuss in this paper extends and complements many related results in literature.

On wavelets Kantorovich (p,q)-Baskakov operators and approximation properties

In this paper, we generalize and extend the Baskakov-Kantorovich operators by constructing the (p, q)-Baskakov Kantorovich operators (ϒn,b,p,qh)(x)=[n]p,q∑b=0∞qb−1υb,np,q(x)∫Rh(y)Ψ([n]p,qqb−1pn−1y−[b]p,q)dp,qy.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\begin{aligned} (\\Upsilon _{n,b,p,q} h) (x) = [ n ]_{p,q} \\sum_{b=0}^{ \\infty}q^{b-1} \\upsilon _{b,n}^{p,q}(x) \\int _{\\mathbb{R}}h(y)\\Psi \\biggl( [ n ] _{p,q} \\frac{q^{b-1}}{p^{n-1}}y - [ b ] _{p,q} \\biggr) \\,d_{p,q}y. \\end{aligned} $$\\end{document} The modified Kantorovich (p, q)-Baskakov operators do not generalize the Kantorovich q-Baskakov operators. Thus, we introduce a new form of this operator. We also introduce the following useful conditions, that is, for any 0 leq b leq omega , such that omega in mathbb{N}, Psi _{omega} is a continuous derivative function, and 0< q< p leq 1, we have int _{mathbb{R}}x^{b}Psi _{omega}(x),d_{p,q}x = 0 . Also, for every Psi in L_{infty},there exists a finite constant γ such that gamma > 0 with the property Psi subset [ 0, gamma ] ,its first ω moment vanishes, that is, for 1 leq b leq omega , we have that int _{mathbb{R}}y^{b}Psi (y),d_{p,q}y = 0,and int _{mathbb{R}}Psi (y),d_{p,q}y = 1. Furthermore, we estimate the moments and norm of the new operators. And finally, we give an upper bound for the operator’s norm.

Gagliardo representation of norms of ball quasi-Banach function spaces

Let X be a ball quasi-Banach function space on Rn. In this article, under some mild assumptions about both X and the boundedness of the Hardy–Littlewood maximal operator on the associate space of the convexification of X, the authors prove that, for any f∈(X/R)∩⋃s∈(0,1)W˙Xs,q(Rn),‖f‖X/R≲lim infs→0+s1q‖[∫Rn|f(⋅)−f(y)|q|⋅−y|n+sqdy]1q‖X≤lim sups→0+s1q‖[∫Rn|f(⋅)−f(y)|q|⋅−y|n+sqdy]1q‖X≲‖f‖X/R and, for any γ∈R∖{0} and f∈X/R,supλ∈(0,∞)⁡λ‖[∫Rn1Ef(λ,q,γ)(⋅,y)|⋅−y|γ−ndy]1q‖X∼‖f‖X/R with the positive equivalence constants independent of f, where f∈X/R if and only if there exists a∈R such that f+a∈X and where ‖f‖X/R:=infa∈R⁡‖f+a‖X<∞, the index q∈(0,∞) is related to X, W˙Xs,q(Rn) is the homogeneous fractional Sobolev space associated with the ball quasi-Banach function space X, andEf(λ,q,γ):={(x,y)∈Rn×Rn:|f(x)−f(y)|>λ|x−y|γq}. In the case when X:=Lp(Rn) with 1≤q=p<∞ and f∈X, the first formula is closely related to the celebrated classical formula of V. Maz'ya and T. Shaposhnikova and the second formula is exactly the recent formula of H. Brezis et al. These results are new even when X=Lp(Rn) with 1≤q<p<∞ and 0<q≤p<1. All these results are of quite wide generality and, even when they are applied to various specific function spaces, most of the obtained results are new. To obtain these results, the authors overcome those obstacles caused by the deficiency of both the translation invariance and an explicit expression of the quasi-norm of X under consideration via establishing some weighted estimates and new decompositions, which depend on the extrapolation and the exact operator norm of the Hardy–Littlewood maximal operator.

Resolvent Convergence for Differential–Difference Operators with Small Variable Translations

We consider general higher-order matrix elliptic differential–difference operators in arbitrary domains with small variable translations in lower-order terms. The operators are introduced by means of general higher-order quadratic forms on arbitrary domains. Each lower-order term depends on its own translation and all translations are governed by a small multi-dimensional parameter. The operators are considered either on the entire space or an arbitrary multi-dimensional domain with a regular boundary. The boundary conditions are also arbitrary and general and involve small variable translations. Our main results state that the considered operators converge in the norm resolvent sense to ones with zero translations in the best possible operator norm. Estimates for the convergence rates are established as well. We also prove the convergence of the spectra and pseudospectra.

On the Minimal Norm of a Projection Operator for Linear Interpolation on an $$n$$-Dimensional Ball

On the Minimal Norm of a Projection Operator for Linear Interpolation on an $$n$$-Dimensional Ball

Criteria for the boundedness of a certain class of matrix operators from lpv into lqu

One of the main aims in the theory of matrices is to find necessary and sufficient conditions for the elements of any matrix so that the corresponding matrix operator maps continuously from one normed space into another one. Thus, it is very important to find the norm of the matrix operator, at least, to find upper and lower estimates of it. This problem in Lebesgue spaces of sequences in the general case is still open. This paper deals with the problem of boundedness of matrix operators from lpv into lqu for 1 &lt; q &lt; p &lt; ∞, and we obtain necessary and sufficient conditions of this problem when matrix operators belong to the classes O2+- satisfying weaker conditions than Oinarov’s condition.

An inertial-type algorithm for a class of bilevel variational inequalities with the split variational inequality problem constraints

In this paper, we investigate the problem of solving strongly monotone variational inequalities over the solution set of the split variational inequality problem with multiple output sets in real Hilbert spaces and establish a new iterative algorithm for it. Our algorithms are accelerated by the inertial technique and eliminate the dependence on the norm of the transformation operators and the strongly monotone and Lipschitz continuous constants of the involved operator by employing a self-adaptive step size criterion. The strong convergence result is given under some mild conditions widely used in the convergence analysis. Two corollaries for the solutions of the split variational inequality problem and the split feasibility problem with multiple output sets are also obtained using our main result. Finally, some numerical experiments have been conducted to illustrate the effectiveness of the proposed algorithms and compare them with the related ones.

Norm-Attainable Operators and Polynomials: Theory, Characterization, and Applications in Optimization and Functional Analysis

This research paper offers a comprehensive investigation into the concept of norm-attainability in Banach and Hilbert spaces. It establishes that norm-attainable operators exist if and only if the target space is a Banach space and that norm-attainable polynomials are inherently linear. In convex optimization scenarios, norm-attainable polynomials lead to unique global optima. The paper explores the norm of norm-attainable operators, revealing its connection to supremum norms. In Hilbert spaces, norm-attainable operators are self-adjoint. Additionally, it shows that in finite-dimensional spaces, all bounded linear operators are norm-attainable. The research also examines extremal polynomials and their relationship with derivative roots, characterizes optimal solutions in norm-attainable operator contexts, and explores equivalence between norm-attainable operators through invertible operators. In inner product spaces, norm-attainable polynomials are identified as constant. Lastly, it highlights the association between norm-attainable operators and convex optimization problems, where solutions lie on the unit ball's boundary. This paper offers a unified perspective with significant implications for functional analysis, operator theory, and optimization in various mathematical and scientific domains.

Inertial Halpern-type method for solving split feasibility and fixed point problems via dynamical stepsize in real Banach spaces

AbstractIn this paper, we introduce a modified Halpern inertial method for approximating solutions of split feasibility problem and fixed point problem of Bregman strongly nonexpansive mappings in the framework of p-uniformly convex and uniformly smooth real Banach spaces. We establish a strong convergence result for the sequence generated by our iterative scheme under some mild conditions without the computation of the operator norm. We state some consequences and present some examples to show the efficiency and implementation of our proposed method. The result discussed in this paper extends and generalizes many recent results in this direction. Our result extends and complements some related results in literature.

Culture of Collaboration: How Madison County Government Drew from its Roots and used Rural Appalachian Values to Increase Government Participation through COVID Service Delivery

This study employs a storytelling approach to describe how a small, rural county in southern Appalachia drew from its roots to combat a pandemic, and in the process increased its resilience. In Madison County, leaders supported their communities by developing cross-sector collaborations that dramatically increased public participation on issues that most affected the county: education access, community outreach, food security and COVID vaccination delivery. Leaders built upon cultural norms and values–honest intent, community spirit, kindness, helping neighbors, hard work, and self-reliance–to increase participation. The results are modeled as an intervention, a flowchart that includes prior county operational norms, cultural norms and values, new models for government operations, and impacts on service delivery. There is a particular focus on mechanisms, the often-unspoken psychological processes that supported the transition in government operations and community and individual participation. Employing storytelling techniques amplifies the explanation of how the processes increased the county’s resilience, and outlines a way of working that they will use to address existing and new complex challenges as they arise.

When are the norms of the Riesz projection and the backward shift operator equal to one?

The lower estimate by Gohberg and Krupnik (1968) and the upper estimate by Hollenbeck and Verbitsky (2000) for the norm of the Riesz projection P on the Lebesgue space Lp lead to ‖P‖Lp→Lp=1/sin⁡(π/p) for every p∈(1,∞). Hence L2 is the only space among all Lebesgue spaces Lp for which the norm of the Riesz projection P is equal to one. Banach function spaces X are far-reaching generalisations of Lebesgue spaces Lp. We prove that the norm of P is equal to one on the space X if and only if X coincides with L2 and there exists a constant C∈(0,∞) such that ‖f‖X=C‖f‖L2 for all functions f∈X. Independently from this, we also show that the norm of P on X is equal to one if and only if the norm of the backward shift operator S on the abstract Hardy space H[X] built upon X is equal to one.

Stability analysis of fully dynamic piezoelectric beams with internal fractional delay

In this paper, we are interested in the stability analysis of one-dimensional fully dynamic piezoelectric beam systems with internal fractional delay under different feedback controls (boundary and internal). By introducing two new equations to deal with fractional delay, equivalent new systems are obtained. Based on classical semigroup theory, we prove the well-posedness of the related systems. In the analysis of their stability, different research methods are adopted in this paper, aiming to reach conclusions in a more concise way. One of the most critical is some estimations which are caused by the existence of fractional delay terms. When the feedback control acts on the boundary, we give the non-trivial conditions of the delay parameter and the control parameter, then the exponential decay of the system with boundary feedback is obtained by using Lyapunov functional theory. When the feedback control operates in the internal domain, the delay parameter is less than the control parameter, the exponential stability of system with internal feedback is obtained by estimating of the resolvent operator norm. This is the first study of piezoelectric beam system with fractional delay and magnetic effect, which has certain significance in its development.

On a linear operator between weighted‐type spaces of analytic functions

We characterize the boundedness, compactness, and estimate essential norm of the polynomial differentiation composition operator between weighted‐type spaces of analytic functions, in terms of the operator symbols and weights.

REGULARIZED JACOBI-TYPE ADMM-METHOD FOR FINDING SOLUTIONS TO GENERALIZED NASH EQUILIBRIUM PROBLEM

In this paper, we extended the well-known alternating direction method of multipliers (ADMM) for optimization problems to generalized Nash equilibrium problems (GNEP) with shared constraints. We developed an ADMM-type algorithm with fixed regularization to tackle the problem (GNEP) where an upper estimate for the operator norm is not known and then we apply a multiplier-penalty in order to get rid of the joint constraints. We equipped the Hilbert space with an appropriate weighted scalar product and it turns out to be weakly convergent under a lipschitz and monotonicity assumption. A proximal term is then added to improve the convergence properties. Furthermore, a comparative analysis of quasi-variational inequality method, interior point method, penalty method and the proposed method are discussed.

A NEW METHOD FOR NORM OF A HARDY-VOLTERRA OPERATOR

In this paper we consider the problem of finding the norm of the Hardy-Volterra operator with convolution kenel in weighted Lebesgue spaces.