Operator Norm Research Articles

On the essential norms of Toeplitz operators on abstract Hardy spaces built upon Banach function spaces

Let X be a Banach function space over the unit circle such that the Riesz projection P is bounded on X and let H[X] be the abstract Hardy space built upon X. We show that the essential norm of the Toeplitz operator T(a):H[X]→H[X]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T(a):H[X]\\rightarrow H[X]$$\\end{document} coincides with ‖a‖L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert a\\Vert _{L^\\infty }$$\\end{document} for every a∈C+H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a\\in C+H^\\infty $$\\end{document} if and only if the essential norm of the backward shift operator T(e-1):H[X]→H[X]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T(\ extbf{e}_{-1}):H[X]\\rightarrow H[X]$$\\end{document} is equal to one, where e-1(z)=z-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{e}_{-1}(z)=z^{-1}$$\\end{document}. This result extends an observation by Böttcher, Krupnik, and Silbermann for the case of classical Hardy spaces.

The pseudospectrum and transient of Kaluza–Klein black holes in Einstein–Gauss–Bonnet gravity

Abstract The spectrum and dynamical instability, as well as the transient effect of the tensor perturbation for the so-called Maeda–Dadhich black hole, a type of Kaluza–Klein black hole, in Einstein–Gauss–Bonnet gravity have been investigated in framework of pseudospectrum. We cast the problem of solving quasinormal modes (QNMs) in AdS-like spacetime as the linear evolution problem of the non-normal operator in null slicing by using ingoing Eddington–Finkelstein coordinates. In terms of spectrum instability, based on the generalized eigenvalue problem, the QNM spectrum and ε-pseudospectrum has been studied, while the open structure of ε-pseudospectrum caused by the non-normality of operator indicates the spectrum instability. In terms of dynamical instability, we introduce the concept of the distance to dynamical instability, which plays a crucial role in bridging the spectrum instability and the dynamical instability. We calculate such distance, named the complex stability radius, as parameters vary. Finally, we show the behavior of the energy norm of the evolution operator, which can be roughly reflected by the three kinds of abscissas in context of pseudospectrum, and find the transient growth of the energy norm of the evolution operator.

Some characterizations of minimal matrices with operator norm

Some characterizations of minimal matrices with operator norm

Necessary and Sufficient Conditions for the Boundedness of Multiple Integral Operators with Super-Homogeneous Kernels in Weighted Lebesgue Space

Super-homogeneous functions including homogeneous functions, quasi-homogeneous functions, and several non-homogeneous functions are considered. Using the weight function method, the construction conditions of Hilbert-type multiple integral inequalities with super-homogeneous kernels are first discussed. Then, using the obtained results, the construction problem of bounded multiple integral operators with super-homogeneous kernels in weighted Lebesgue space is discussed, and the necessary and sufficient conditions for operator boundedness and the operator norm formula are obtained.

Redefining organizational digitality: a relational-ontological approach inspired by new materialism

Organizations, as central actors in societal structure, undergo significant transformations due to the impact of digitalization, often resulting in disruptive changes. Consequently, organizations increasingly view digitalization as an ongoing process of negotiation, which has led to the emergence of new operational modes and organizational norms. In this context, the interaction between organizations and digital technologies is characterized by recursive dynamics, which blur conventional boundaries. This presents a challenge in defining the distinct domains of the digital and the organizational within the framework of recursivity. This article draws upon new materialism and agential realism to propose an ontological-relational approach to understanding organizational digitality. This approach suggests a reconceptualization of organizational digitality as a mechanism that generates relational entities, thereby reshaping their inherent meanings. By transcending traditional boundaries between organizations and digital, this perspective provides a nuanced understanding of digital phenomena within organizational contexts.

Operator ℓp → ℓq norms of random matrices with iid entries

We prove that for every p,q∈[1,∞] and every random matrix X=(Xi,j)i≤m,j≤n with iid centered entries satisfying the α-regularity assumption ‖Xi,j‖2ρ≤α‖Xi,j‖ρ for every ρ≥1, the expectation of the operator norm of X from ℓpn to ℓqm is comparable, up to a constant depending only on α, tom1/qsupt∈Bpn⁡‖∑j=1ntjX1,j‖q∧Logm+n1/p⁎sups∈Bq⁎m⁡‖∑i=1msiXi,1‖p⁎∧Logn. We give more explicit formulas, expressed as exact functions of p, q, m, and n, for the two-sided bounds of the operator norms in the case when the entries Xi,j are: Gaussian, Weibullian, log-concave tailed, and log-convex tailed. In the range 1≤q≤2≤p we provide two-sided bounds under the weaker regularity assumption (EX1,14)1/4≤α(EX1,12)1/2.

Discrepancies in Euclidean Operator Radii in Hilbert C∗-Modules

In this research, we establish precise limits for the Euclidean operator radius of two bounded linear operators operating within a Hilbert C∗-module over A. Furthermore, our work establishes a connection between these limits and recent research findings that provide accurate upper and lower bounds for the numerical radius of linear operators. The primary objective of this investigation is to explore various specific scenarios of interest and extend existing inequalities found in the literature to encompass the Euclidean radius of two operators in a Hilbert A-module. Additionally, our study presents conclusions that reveal relationships between the operator norm, the typical numerical radius of a composite operator, and the Euclidean operator radius. Furthermore, we introduce several new inequalities involving the Euclidean numerical radius and Euclidean operator norm of 2-tuple operators. These inequalities offer both lower and upper bounds for the Euclidean numerical radius of 2-tuple operators, as well as for the sum and product of 2-tuple operators. We also delve into the study of Euclidean numerical radius inequalities for 2×2 operator matrices whose entries consist of 2-tuple operators, leading to the derivation of some Euclidean operator radius inequalities. Additionally, we establish an inequality for the Euclidean operator norm of 2×2 operator matrices.

Weighted Rellich and Hardy inequalities in L p(·) spaces

Abstract In this note we prove weighted Rellich–Sobolev and Hardy–Sobolev inequalities in variable exponent Lebesgue spaces L p ⁢ ( ⋅ ) ⁢ ( 𝔾 ) {L^{p(\,\cdot\,)}(\mathbb{G})} defined on stratified homogeneous groups 𝔾 {\mathbb{G}} . To derive the main results, we rely on weighted estimates for the Riesz potential operators in L p ⁢ ( ⋅ ) ⁢ ( 𝔾 ) {L^{p(\,\cdot\,)}(\mathbb{G})} , where 𝔾 {{\mathbb{G}}} is a general homogeneous group. The results are new even for the Abelian (Euclidean) case 𝔾 = ( ℝ d , + ) {\mathbb{G}=(\mathbb{R}^{d},+)} and the Heisenberg groups 𝔾 = ℍ n {\mathbb{G}={\mathbb{H}}^{n}} . The main statements are obtained for variable exponents satisfying the condition that the Hardy–Littlewood maximal operator is bounded in appropriate variable exponent Lebesgue spaces. We also give some quantitative estimates for the norms of integral operators involved in derived estimates.

Higher-order Lp isoperimetric and Sobolev inequalities

Schneider introduced an inter-dimensional difference body operator on convex bodies, and proved an associated inequality. In the prequel to this work, we showed that this concept can be extended to a rich class of operators from convex geometry and proved the associated isoperimetric inequalities. The role of cosine-like operators, which generate convex bodies in Rn from those in Rn, were replaced by inter-dimensional simplicial operators, which generate convex bodies in Rnm from those in Rn (or vice versa). In this work, we treat the Lp extensions of these operators, and, furthermore, extend the role of the simplex to arbitrary m-dimensional convex bodies containing the origin. We establish mth-order Lp isoperimetric inequalities, including the mth-order versions of the Lp Petty projection inequality, Lp Busemann-Petty centroid inequality, Lp Santaló inequalities, and Lp affine Sobolev inequalities. As an application, we obtain isoperimetric inequalities for the volume of the operator norm of linear functionals (Rn,‖⋅‖E)→(Rm,‖⋅‖F).

The equivalent conditions for norm of a Hilbert-type integral operator with a combination kernel and its applications

Introducing adaptation parameters σ,σ1, formal parameters λi(i=1,2,3,4),κ,τ, and type parameters μ,ν, the integration operator is defined as T:Lpp(1−μσˆ)−1(R+)→Lppνσˆ−1(R+), Tf(y)=∫R+eλ1xμyν+κe−λ2xμyνeλ3xμyν+τe−λ4xμyνf(x)dx,y∈R+. Using the weight function method, a general Hilbert-type integral inequality is obtained, thereby proving the boundedness of the operator. The constant factor of the general Hilbert-type inequality is the best possible if and only if the adaptation parameters satisfy σ=σ1. From this, the formula for calculating the operator norm is obtained. In terms of application, some results from the references have been consolidated by discussing the combination of formal parameters and type parameters, and many new operator inequalities of different types and forms have been derived.

Norm-Peak Multilinear Forms on the Plane with the Rotated Supremum Norm

Let 𝑛 ≥ 2. A continuous 𝑛-linear form 𝑇 on a Banach space 𝐸 is called norm-peak if there is a unique (𝑥1, … , 𝑥𝑛) ∈ 𝐸𝑛 such that ║𝑥1║ = … = ║𝑥𝑛║ = 1 and for the multilinear operator norm it holds ‖𝑇 ‖ = |𝑇 (𝑥1, … , 𝑥𝑛)|.Let 0 ≤ 𝜃 ≤ = ℝ2 with the rotated supremum norm ‖(𝑥, 𝑦)‖(∞,𝜃) = max {|𝑥 cos 𝜃 + 𝑦 sin 𝜃|, |𝑥 sin 𝜃 − 𝑦 cos 𝜃|}.In this note, we characterize all norm-peak multilinear forms on . As a corollary we characterize all norm-peak multilinear forms on = ℝ2 with the 𝓁𝑝-norm for 𝑝 = 1, ∞.

Novel Accelerated Cyclic Iterative Approximation for Hierarchical Variational Inequalities Constrained by Multiple-Set Split Common Fixed-Point Problems

In this paper, we investigate a class of hierarchical variational inequalities (HVIPs, i.e., strongly monotone variational inequality problems defined on the solution set of multiple-set split common fixed-point problems) with quasi-pseudocontractive mappings in real Hilbert spaces, with special cases being able to be found in many important engineering practical applications, such as image recognizing, signal processing, and machine learning. In order to solve HVIPs of potential application value, inspired by the primal-dual algorithm, we propose a novel accelerated cyclic iterative algorithm that combines the inertial method with a correction term and a self-adaptive step-size technique. Our approach eliminates the need for prior knowledge of the bounded linear operator norm. Under appropriate assumptions, we establish strong convergence of the algorithm. Finally, we apply our novel iterative approximation to solve multiple-set split feasibility problems and verify the effectiveness of the proposed iterative algorithm through numerical results.

Gluonic evanescent operators: negative-norm states and complex anomalous dimensions

In this paper, we build on our previous work to further investigate the role of evanescent operators in gauge theories, with a particular focus on their contribution to violations of unitarity. We develop an efficient method for calculating the norms of gauge-invariant operators in Yang-Mills (YM) theory by employing on-shell form factors. Our analysis, applicable to general spacetime dimensions, reveals the existence of negative-norm states among evanescent operators. We also explore the one-loop anomalous dimensions of these operators and find complex anomalous dimensions. We broaden our analysis by considering YM theory coupled with scalar fields and we observe similar patterns of non-unitarity. The presence of negative-norm states and complex anomalous dimensions across these analyses provides compelling evidence that general gauge theories are non-unitary in non-integer spacetime dimensions.

Modi ed inertia Halpern method for split null point problem in Banach spaces

Abstract. In this paper, we study split null point problem in reflexive Banach spaces. Using the Bregman technique together with a modified inertial Halpern method, we approximate a solution of split null point problem. Also, we establish a strong convergence result for approximating the solution of the aforementioned problems. It is worth mentioning that the iterative algorithm employ in this study is design in such a way that it does not require prior knowledge of operator norm. We display some numerical examples to illustrate the performance of the proposed iterative method. The result discuss in this paper extends and complements many related results in literature. Mathematics Subject Classification (2010): 47H06, 47H09, 47J05, 47J25. Keywords: Monotone variational inclusion problem, split feasibility problem, firmly nonexpansive-type mapping, fixed point problem, inertial method.

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.

General Upper Bounds for the Numerical Radii of Hilbert Space Operators

We present a collection upper bounds for the numerical radii of a certain 2 × 2 operator matrices. We use these bounds to improve on some known numerical radius inequalities for powers of Hilbert space operators. In particular, we show that if 𝐴 is a bounded linear operator on a complex Hilbert space, then 𝑤 2𝑟 (𝐴) ≤ 1+𝛼 8 ‖|𝐴| 2𝑟 +|𝐴 ∗ | 2𝑟‖+ 1+𝛼 4 𝑤(|𝐴| 𝑟 |𝐴 ∗ | 𝑟 )+ 1−𝛼 2 𝑤 𝑟 (𝐴 2 ) for every r ≥ 1 and α ∈ [0,1]. This substantially improves on the existing inequality 𝑤 2𝑟 (𝐴) ≤ 1 2 ‖|𝐴| 2𝑟 + |𝐴 ∗ | 2𝑟‖. Here 𝑤(. ) and ||. || denote the numerical radius and the usual operator norm, respectively.

Spherical p-quasinorms of operator tuples

In this paper, we introduce the notion of the spherical p-quasinorms for operator tuples, and study some of their properties. We prove several inequalities and relations involving the different types of joint operator norms and spherical p-quasinorms, giving refinements of some fundamental inequalities, along the way. Among other things, we prove that ‖T‖Q,p=ωp(T), for any p>1 and any range-orthogonal normal tuple T. Finally, we also consider the asymptotic behaviour of p-quasinorms when p→0+ and use it to derive some bounds for the product of commuting normal operators.

An approximate decoupled reliability-based design optimization method for efficient design exploration of linear structures under random loads

Reliability-based design optimization (RBDO) provides a promising approach for achieving effective structural designs while explicitly accounting for the effects of uncertainty. However, the computational demands associated with RBDO, often due to its nested loop nature, pose significant challenges, thereby impeding the application of RBDO for decision-making in real-world structural design. To alleviate this issue, an approximate decoupled approach is introduced for a class of RBDO problems involving linear truss structures subjected to random excitations, with the failure event defined by compliance. This contribution aims to provide an approximate but efficient way for design exploration to facilitate decision-making during the initial design phase. Specifically, the proposed approach converts the original RBDO problem into a deterministic optimization problem through a modest number of reliability analyses by the probability density evolution method (PDEM). Once the deterministic optimization problem is obtained, the solution of the whole RBDO problem can be obtained by solving this equivalent problem without further reliability analysis, resulting in notable enhancement in terms of computational efficiency. In this way, this contribution expands the frontier of application of the operator norm theory within the RBDO framework. Numerical examples are conducted to illustrate the effectiveness and capabilities of the proposed approach.

A NEW CONVERGENCE ANALYSIS FOR MINIMUM NORM SOLUTION OF SPLIT SYSTEM OF NONSMOOTH MINIMIZATION PROBLEMS

This work presents a novel self-adaptive steepest-descent type algorithm for solving the split system of minimization problem (SSMP) related to convex nonsmooth functions. The algorithm includes a self-adaptive step size mechanism, which uses a step size that does not need prior information about the operator norm. Under certain weakened assumptions of parameters, a strong convergence theorem is established and proved for the algorithm. Specifically, the sequence generated by this new algorithm strongly converges towards the minimum norm element of the SSPM. To assess the implementation of our algorithm, a numerical example is provided. According to the numerical results, our algorithm showcases effectiveness and simplicity in its implementation. Furthermore, the primary numerical experiment results indicate that our proposed algorithm surpasses some existing results in the literature in terms of CPU time and iteration count. Our result represents an extension and enhancement of recent findings in this area.

Invariant tensions from holography

We revisit the problem of defining an invariant notion of tension in gravity. For spacetimes whose asymptotics are those of a Defect CFT we propose two independent definitions: gravitational tension given by the one-point function of the dilatation current, and inertial tension, or stiffness, given by the norm of the displacement operator. We show that both reduce to the tension of the Nambu-Goto action in the limit of classical thin-brane probes. Subtle normalisations of the relevant Witten diagrams are fixed by the Weyl and diffeomorphism Ward identities of the boundary DCFT. The gravitational tension is not defined for domain walls, whereas stiffness is not defined for point particles. When they both exist these two tensions are in general different, but the examples of line and surface BPS defects in d = 4 show that superconformal invariance can identify them.