Operator Norm Research Articles

On split generalized equilibrium problem with multiple output sets and common fixed points problem

Abstract In this article, we introduce and study the notion of split generalized equilibrium problem with multiple output sets (SGEPMOS). We propose a new iterative method that employs viscosity approximation technique for approximating the common solution of the SGEPMOS and common fixed point problem for an infinite family of multivalued demicontractive mappings in real Hilbert spaces. Under mild conditions, we prove a strong convergence theorem for the proposed method. Our method uses self-adaptive step size that does not require prior knowledge of the operator norm. The results presented in this article unify, complement, and extend several existing recent results in the literature.

Two weight inequality for Hankel form on weighted Bergman spaces induced by doubling weights

The boundedness of the small Hankel operator hfν(g)=Pν(fg‾), induced by an analytic symbol f and the Bergman projection Pν associated to ν, acting from the weighted Bergman space Aωp to Aνq is characterized on the full range 0<p,q<∞ of parameters when ω and ν belong to the class D of radial weights admitting certain two-sided doubling conditions. Moreover, an asymptotic formula for the operator norm of hfν is established in terms of a suitable norm of f(n) depending upon the inducing weights and parameters. Certain results obtained are equivalent to the boundedness of bilinear Hankel forms, which are in turn used to establish the weak factorization Aηq=Aωp1⊙Aνp2, where 1<q,p1,p2<∞ such that q−1=p1−1+p2−1 and η˜1q≍ω˜1p1ν˜1p2. Here τ˜(r)=∫r1τ(t)dt/(1−r) for all 0≤r<1.

Extrapolation in general quasi-Banach function spaces

In this work we prove off-diagonal, limited range, multilinear, vector-valued, and two-weight versions of the Rubio de Francia extrapolation theorem in general quasi-Banach function spaces. We prove mapping properties of the generalization of the Hardy-Littlewood maximal operator to very general bases that includes a method to obtain self-improvement results that are sharp with respect to its operator norm. Furthermore, we prove bounds for the Hardy-Littlewood maximal operator in weighted Lorentz, variable Lebesgue, and Morrey spaces, and recover and extend several extrapolation theorems in the literature. Finally, we provide an application of our results to the Riesz potential and the Bilinear Hilbert transform.

Hierarchical variational inequality problem and split common fixed point of averaged operators

In this paper, we consider a variational inequality problem which is defined over the intersection of the solution set of variational inequality problem and the solution set of split common fixed point problem for averaged operators in Hilbert spaces. We present a new and efficient iterative method for solving this problem and establish its strong convergence. The proposed algorithm does not require knowledge of the Lipschitz constants of the operators and does not depend on the prior information of the bounded linear operator norms. Finally, we apply our result to study certain classes of optimization problems, and present some numerical experiments to demonstrate the applicability of our proposed algorithm.

Gains of integrability and local smoothing effects for quadratic evolution equations

We characterize geometrically the semigroups generated by non-selfadjoint quadratic differential operators (e−tqw)t≥0 enjoying local smoothing effects and providing gains of integrability. More precisely, we prove that the evolution operators e−tqw map Lp on Lq∩C∞, for all 1≤p≤q≤∞, if and only if the singular space of the quadratic operator qw is included in the graph of a linear map. We also provide quantitative estimates for the associated operator norms in the short-time asymptotics 0<t≪1.

A kind of sharp Wirtinger inequalities

This study gives a kind of sharp Wirtinger inequalities (Pizone inequalities) [Formula: see text] where [Formula: see text] with at least [Formula: see text] zeros (counting multiplicity) in [Formula: see text]. First, based on the Hermite (Lagrange) interpolation, we express [Formula: see text] as a Lagrange type (integral type) remainder. Second, we refer the computation of [Formula: see text] to the maximum value problem of a multivariate function, and we give the values of [Formula: see text] by finding the solution of the multivariate function aforementioned. At last, we refer the computation of [Formula: see text] to the norm of an integral operator. Our results are corrections and extensions to the results that appear in [J. C. Kuang, Applied Inequalities (Shandong Science and Technology Press, Jinan, 2004); A. Yu. Levin, Some estimates for a diff erentiable function, Dokl. Akad. Nauk SSSR 138 (1961) 37-38 (in Russian)].

Improvement of Menshov’s Theorem on Upper Estimate for the Norm of Maximal Operator

Improvement of Menshov’s Theorem on Upper Estimate for the Norm of Maximal Operator

Spectra of composition operators on the Dirichlet space induced by full-maps of the disk

Motivated by the study of composition operators induced by inner functions acting on the Hardy space, we present spectral results for composition operators induced by a class of full-maps of the disk when acting on the Dirichlet space. To do this, we first investigate the norm, essential norm, spectral radius, and essential spectral radius of a composition operator when the inducing map has bounded valence. Our spectral results consider composition operators induced by univalent full-maps, monomials, finite Blaschke products, and general full-maps of the disk with valence that is constant almost everywhere.

Growth bound and threshold dynamic for nonautonomous nondensely defined evolution problems.

We propose a general framework for simultaneously calculating the threshold value for population growth and determining the sign of the growth bound of the evolution family generated by the problem below [Formula: see text]where [Formula: see text] is a Hille-Yosida linear operator (possibly unbounded, non-densely defined) on a Banach space [Formula: see text], and the maps [Formula: see text], [Formula: see text] are p-periodic in time and continuous in the operator norm topology. We give applications of our approach for two general examples of an age-structured model, and a delay differential system. Other examples concern the dynamics of a nonlocal problem arising in population genetics and the dynamics of a structured human-vector malaria model.

Limiting spectral distribution of stochastic block model

The stochastic block model (SBM) is an extension of the Erdős–Rényi graph and has applications in numerous fields, such as data analysis, recovering community structure in graph data and social networks. In this paper, we consider the normal central SBM adjacency matrix with [Formula: see text] communities of arbitrary sizes. We derive an explicit formula for the limiting empirical spectral density function when the size of the matrix tends to infinity. We also obtain an upper bound for the operator norm of such random matrices by means of the Stieltjes transform and random matrix theory.

Q-Numerical radius inequalities for Hilbert space

The aim of this paper is to study the q-numerical radius ω q ( . ) of bounded linear operators on Hilbert spaces. More precisely, first, we show that ω q ( . ) defines a norm which is equivalent to the operator norm. Next, the following compatible generalization of Kittaneh's inequality 1 4 ( q 2 − q 2 ) 2 ∥ T ∗ T + T T ∗ ∥ ≤ ω q 2 ( T ) ≤ ( q + 2 1 − q 2 ) 2 2 × ‖ T ∗ T + T T ∗ ‖ . is obtained. Finally, some generalizations of q-numerical radius inequalities for composition of operators are established.

The norm of time-frequency and wavelet localization operators

Time-frequency localization operators (with Gaussian window) L F : L 2 ( R d ) → L 2 ( R d ) L_F:L^2(\mathbb {R}^d)\to L^2(\mathbb {R}^d) , where F F is a weight in R 2 d \mathbb {R}^{2d} , were introduced in signal processing by I. Daubechies [IEEE Trans. Inform. Theory 34 (1988), pp. 605–612], inaugurating a new, geometric, phase-space perspective. Sharp upper bounds for the norm (and the singular values) of such operators turn out to be a challenging issue with deep applications in signal recovery, quantum physics and the study of uncertainty principles. In this note we provide optimal upper bounds for the operator norm ‖ L F ‖ L 2 → L 2 \|L_F\|_{L^2\to L^2} , assuming F ∈ L p ( R 2 d ) F\in L^p(\mathbb {R}^{2d}) , 1 &gt; p &gt; ∞ 1&gt;p&gt;\infty or F ∈ L p ( R 2 d ) ∩ L ∞ ( R 2 d ) F\in L^p(\mathbb {R}^{2d})\cap L^\infty (\mathbb {R}^{2d}) , 1 ≤ p &gt; ∞ 1\leq p&gt;\infty . It turns out that two regimes arise, depending on whether the quantity ‖ F ‖ L p / ‖ F ‖ L ∞ \|F\|_{L^p}/\|F\|_{L^\infty } is less or greater than a certain critical value. In the first regime the extremal weights F F , for which equality occurs in the estimates, are certain Gaussians, whereas in the second regime they are proved to be Gaussians truncated above, degenerating into a multiple of a characteristic function of a ball for p = 1 p=1 . This phase transition through Gaussians truncated above appears to be a new phenomenon in time-frequency concentration problems. For the analogous problem for wavelet localization operators—where the Cauchy wavelet plays the role of the above Gaussian window—a complete solution is also provided.

Transference and restriction of Fourier multipliers on Orlicz spaces

AbstractLet G be a locally compact abelian group with Haar measure and be Young functions. A bounded measurable function m on G is called a Fourier ‐multiplier if defined for functions in such that , extends to a bounded operator from to . We write for the space of ‐multipliers on G and study some properties of this class. We give necessary and sufficient conditions for m to be a ‐multiplier on various groups such as , and . In particular, we prove that regulated ‐multipliers defined on coincide with ‐multipliers defined on the real line with the discrete topology D, under certain assumptions involving the norm of the dilation operator acting on Orlicz spaces. Also, several transference and restriction results on multipliers acting on and are achieved.

Upper bounds for the numerical radii of powers of Hilbert space operators

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

Minimal self-adjoint compact operators, moment of a subspace and joint numerical range

We define the (convex) joint numerical range for an infinite family of compact operators in a Hilbert space H. We use this set to determine whether a self-adjoint compact operator A with ±‖A‖ in its spectrum is minimal respect to the set of diagonals in a fixed basis E of H in the operator norm, that is ‖A‖≤‖A+D‖, for all diagonal D. We also describe the moment set mS=conv{|v|2:v∈S and ‖v‖=1} of a subspace S⊂H in terms of joint numerical ranges and obtain equivalences between the intersection of moments of two subspaces and of its two related joint numerical ranges. Moreover, we relate the condition of minimality of A or the intersection of the moments of the eigenspaces of ±‖A‖ to the intersection of the joint numerical ranges of two finite families of certain finite hermitian matrices. We also study geometric properties of the set mS such as extremal curves related with the basis E. All these conditions are directly related with the description of minimal self-adjoint compact operators.

Chinese emergency expert consensus on bedside temporary cardiac pacing (2023)

Temporary cardiac pacing is an essential technique in the diagnosis and treatment of arrhythmias. Due to its urgency, complexity, and uncertainty, it is necessary to develop an evidence-based emergency operation norms. Currently, there is no specific consensus guidelines at home or abroad. The Emergency Branch of Chinese Medical Association organized relevant experts to draft the Chinese emergency expert consensus on bedside temporary cardiac pacing (2023) to guide the operation and application of bedside cardiac pacing. The formulation of the consensus adopts the consensus meeting method and the evidentiary basis and recommendation grading of the Oxford Center for Evidence-based Medicine in the United States. A total of 13 recommendations were extracted from the discussion on the methods of bedside temporary cardiac pacing, the puncture site of transvenous temporary cardiac pacing, the selection of leads, the placement and placement of leads, pacemaker parameter settings, indications, complications and postoperative management. The recommended consensus includes the choice between transcutaneous and transvenous pacing, preferred venous access for temporary transvenous pacing, the target and best guidance method for implantation of bedside pacing electrodes, recommended default pacemaker settings, recommended indications for sinoatrial node dysfunction, atrioventricular block, acute myocardial infarction, cardiac arrest, ventricular and supraventricular arrhythmias. They also recommended ultrasound guidance and a shortened temporary pacing support time to reduce complications of temporary transvenous cardiac pacing, recommended bedrest, and anticoagulation after temporary transvenous pacing. Bedside temporary cardiac pacing is generally safe and effective. Accurate assessment, correct selection of the pacing mode, and timely performance of bedside temporary cardiac pacing can further improve the survival rate and prognosis of related emergency patients.

Essential norm of intrinsic operators from Banach spaces of analytic functions into weighted-type spaces

Essential norm of intrinsic operators from Banach spaces of analytic functions into weighted-type spaces

Characterisations for uniform amenability

In this paper, we provide several characterisations for uniform amenability concerning a family of finitely generated groups. More precisely, we show that the Hulanicki–Reiter condition for uniform amenability can be weakened in several directions, including cardinalities of supports and certain operator norms.

Strong convergence results of split equality hierarchical fixed point problems

The primary objective of this study is to approximate a nontrivial solution to a significant extension of the hierarchical fixed point problem, known as the split equality hierarchical fixed point problem, in the context of real Hilbert spaces. We present an accelerated simultaneous Halpern‐type iterative algorithm with a step size independent of the operator norms. This method combines accelerated extrapolation and the Halpern‐type iterative algorithm. Moreover, we analyze that the sequence pair induced by the suggested algorithm strongly converges to some solution to the split equality hierarchical fixed point problem under some simple control sequence conditions. Furthermore, we provide numerical experiments to demonstrate the rationality and superiority of our algorithm. Some notable recent findings in this field have been improved, generalized, and extended as an outcome of this paper.

The plasmon–polariton scattering by random-impedance surface defects: The interplay between localization and outflow

We study the scattering of TM-polarized surface plasmon-polariton (SPP) by the finite section of flat metal–vacuum interface with a random impedance. We analyze the solution to the integral equation that connect the scattered field and the incident plasmon–polariton, and is valid for any strength of the scattering and dissipative characteristics of the conducting half-space. We show that the norm of the intermode scattering operator as a measure of scattering strength is not only determined by the parameters of the random impedance (the variance, correlation radius, the length of the heterogeneous section of the interface), but also crucially depends on the metal conductivity. For a small norm of the integral operator, the incident surface plasmon polariton radiates effectively into vacuum, resulting in excitation of quasi-isotropic Norton-type waves above the conducting surface. The intensity of the leaking field is expressed in terms of the pair correlation function of the impedance, whose dependence on wave numbers of incident and scattered waves demonstrates the possibility to observe a phenomenon similar to Wood’s anomalies of wave scattering by periodic gratings. Under strong scattering regime, the radiation into the upper half-space is highly suppressed and the SPP wave is mainly backscattered from the heterogeneous surface segment. For the lossless conducting half-space, the surface plasmon–polariton becomes unstable for arbitrarily small fluctuations of the conductor polarizability. The mirroring should also take place at small norm of the scattering operator, yet in this case it is related to Anderson’s localization of the SPP within the disordered section.