General Class Of Operators Research Articles

Bloom weighted bounds for sparse forms associated to commutators

In this paper we consider bilinear sparse forms intimately related to iterated commutators of a rather general class of operators. We establish Bloom weighted estimates for these forms in the full range of exponents, both in the diagonal and off-diagonal cases. As an application, we obtain new Bloom bounds for commutators of (maximal) rough homogeneous singular integrals and the Bochner–Riesz operator at the critical index. We also raise the question about the sharpness of our estimates. In particular we obtain the surprising fact that even in the case of Calderón–Zygmund operators, the previously known quantitative Bloom bounds are not sharp for the second and higher order commutators.

On some fixed point theorems for Ćirić operators

In this paper, we will present existence and localization fixed point theorems and stability results for the fixed point problem involving some very general classes of operators (single-valued and multi-valued), namely for Ćirić type operators. Then, an application to homotopy principles is given. Our results complement and extend the works in the literature.

Uncertainty relation for non-Hermitian systems

We construct an uncertainty relation for arbitrary finite-dimensional $\mathcal{PT}$-invariant non-Hermitian quantum systems within a special inner product framework. This construction is led by ``good observables,'' which are a more general class of operators. We show that the cumulative gain in the quantum Fisher information when measuring two good observables for such non-Hermitian systems is much better than their Hermitian counterpart. Minimum uncertainty states being the best candidates for this gain near the exceptional point supports the intelligent or simultaneous non-Hermitian quantum sensors.

Mellin transform for fractional integrals with general analytic kernel

<abstract><p>Many different operators of fractional calculus have been proposed, which can be organized in some general classes of operators. According to this study, the class of fractional integrals and derivatives can be classified into two main categories, that is, with and without general analytical kernel (introduced in 2019). In this article, we define the Mellin transform for fractional differential operator with general analytic kernel in both Riemann-Liouville and Caputo derivatives of order $ \varsigma\ge0 $ and $ \varrho $ be a fixed parameter. We will also establish relation between Mellin transform with Laplace and Fourier transforms.</p></abstract>

On fractional calculus with analytic kernels with respect to functions

Many different types of fractional calculus have been proposed, which can be organised into some general classes of operators. For a unified mathematical theory, results should be proved in the most general possible setting. Two important classes of fractional-calculus operators are the fractional integrals and derivatives with respect to functions (dating back to the 1970s) and those with general analytic kernels (introduced in 2019). To cover both of these settings in a single study, we can consider fractional integrals and derivatives with analytic kernels with respect to functions, which have never been studied in detail before. Here we establish the basic properties of these general operators, including series formulae, composition relations, function spaces, and Laplace transforms. The tools of convergent series, from fractional calculus with analytic kernels, and of operational calculus, from fractional calculus with respect to functions, are essential ingredients in the analysis of the general class that covers both.

On the proximal point algorithm and its Halpern-type variant for generalized monotone operators in Hilbert space

In a recent paper, Bauschke et al. study rho -comonotonicity as a generalized notion of monotonicity of set-valued operators A in Hilbert space and characterize this condition on A in terms of the averagedness of its resolvent J_A. In this note we show that this result makes it possible to adapt many proofs of properties of the proximal point algorithm PPA and its strongly convergent Halpern-type variant HPPA to this more general class of operators. This also applies to quantitative results on the rates of convergence or metastability (in the sense of T. Tao). E.g. using this approach we get a simple proof for the convergence of the PPA in the boundedly compact case for rho -comonotone operators and obtain an effective rate of metastability. If A has a modulus of regularity w.r.t. zer, A we also get a rate of convergence to some zero of A even without any compactness assumption. We also study a Halpern-type variant HPPA of the PPA for rho -comonotone operators, prove its strong convergence (without any compactness or regularity assumption) and give a rate of metastability.

Continuity of solutions for the Δϕ-Laplacian operator

In this paper we give sufficient conditions to obtain continuity results of solutions for the so called ϕ-Laplacian Δϕ with respect to domain perturbations. We point out that this kind of results can be extended to a more general class of operators including, for instance, nonlocal nonstandard growth type operators.

Hidden analytic relations for two-loop Higgs amplitudes in QCD

We compute the Higgs plus two-quark and one-gluon amplitudes () and Higgs plus three-gluon amplitudes () in the Higgs effective theory with a general class of operators. By changing the quadratic Casimir C F to C A , the maximally transcendental parts of the amplitudes turn out to be equivalent to that of the amplitudes, which also coincide with the counterparts in SYM. This generalizes the so-called maximal transcendentality principle to the Higgs amplitudes with external quark states, thus the full QCD theory. We further verify that the correspondence applies also to two-loop form factors of more general operators, in both QCD and scalar-YM theory. Another interesting relation is also observed between the planar amplitudes and the minimal density form factors in SYM.

Perturbation theory, M-essential spectra of operator matrices

In this paper, we will establish some results on perturbation theory of block operator matrices acting on Xn, where X is a Banach space. These results are exploited to investigate the M-essential spectra of a general class of operators defined by a 3x3 block operator matrix acting on a product of Banach spaces X3.

Controllability and observability for non-autonomous evolution equations: The averaged Hautus test

We consider the observability problem for non-autonomous evolution systems (i.e., the operators governing the system depend on time). We introduce an averaged Hautus condition and prove that for skew-adjoint operators it characterizes exact observability. Next, we extend this to more general class of operators under a growth condition on the associated evolution family. We give an application to the Schrödinger equation with time dependent potential and the damped wave equation with a time dependent damping coefficient.

Approximation of Entropy Numbers

The purpose of this article is to develop a technique to estimate certain bounds for entropy numbers of diagonal operator on \(\ell ^p\) spaces for \(1<p<\infty ,\) which improves the existing bounds. The approximation method we develop in this direction works for a very general class of operators between Banach spaces, in particular, separable Hilbert spaces. As a consequence of this technique we also obtain an alternative proof for the following result for a bounded linear operator T between two separable Hilbert spaces: $$\begin{aligned} \epsilon _k(T)=\epsilon _k(T^*)=\epsilon _k(|T|) \; \text {for each}\; k\in \mathbb N, \end{aligned}$$where \(\epsilon _k(T)\) is the kth entropy number of T.

Some Property of Finite Sum of Weighted Composition and Weighted Frobenius-Perron Operators

The study of finite sum of weighted composition operators on Lp - spaces has received considerable attention in 2012. Characterizations, basic properties of this operators have been obtained. Weighted composition operators are a general class of operators and they appear naturally in the study of surjective isometries on most of the function spaces, semi grup theory, dynamic systems etc. This type of operators are a generalization of multiplication operators and composition operators. In this paper the relations among completely continuous and M‎ - weakly compact of finite sum of weighted composition operators between Lp(μ) -‎ spaces described, we also obtain some necessary and sufficient conditions for Fredholmness of the finite sum of weighted composition operators.

Szász–Mirakyan Type Operators Which Fix Exponentials

In this paper, we construct a new general class of operators which have the classical Szasz Mirakyan ones as a basis, and fix the functions \(e^{ax}\) and \(e^{2ax}\) with \(a>0\). The convergence of the corresponding sequences is discussed in exponential weighted spaces, and a Voronovskaya type result is given. Also we define a new weighted modulus of smoothness and determine the approximation order of the constructed operators. Finally, we study the goodness of the estimates of our new operators via saturation results.

Partial isometries and the conjecture of C.K. Fong and S.K. Tsui

We investigate some bounded linear operators T on a Hilbert space which satisfy the condition |T|≤|ReT|. We describe the maximum invariant subspace for a contraction T on which T is a partial isometry to obtain that, in certain cases, the above condition ensures that T is self-adjoint. In other words we show that the Fong–Tsui conjecture holds for partial isometries, contractive quasi-isometries, or 2-quasi-isometries, and Brownian isometries of positive covariance, or even for a more general class of operators.

A NOTE ON ∗-PARANORMAL OPERATORS AND RELATED CLASSES OF OPERATORS

We shall show that the Riesz idempotent <TEX>$E_{\lambda}$</TEX> of every *-paranormal operator T on a complex Hilbert space H with respect to each isolated point <TEX>${\lambda}$</TEX> of its spectrum <TEX>${\sigma}(T)$</TEX> is self-adjoint and satisfies <TEX>$E_{\lambda}\mathcal{H}=ker(T-{\lambda})= ker(T-{\lambda})^*$</TEX>. Moreover, Weyl's theorem holds for *-paranormal operators and more general for operators T satisfying the norm condition <TEX>$||Tx||^n{\leq}||T^nx||\,||x||^{n-1}$</TEX> for all <TEX>$x{\in}\mathcal{H}$</TEX>. Finally, for this more general class of operators we find a sufficient condition such that <TEX>$E_{\lambda}\mathcal{H}=ker(T-{\lambda})= ker(T-{\lambda})^*$</TEX> holds.

Some convergence results for iterative sequences of Prešić type and applications

Abstract In this article, we study the convergence of iterative sequences of Prešić type involving new general classes of operators in the setting of metric spaces. As application, we derive some convergence results for a class of nonlinear matrix difference equations. Numerical experiments are also presented to illustrate the convergence algorithms. Mathematics Subject Classification 2000: 54H25; 47H10; 15A24; 65H05.

Generalizations of the classical Yang-Baxter equation and ${\mathcal O}$O-operators

Tensor solutions (r-matrices) of the classical Yang-Baxter equation (CYBE) in a Lie algebra, obtained as the classical limit of the R-matrix solution of the quantum Yang-Baxter equation, is an important structure appearing in different areas such as integrable systems, symplectic geometry, quantum groups, and quantum field theory. Further study of CYBE led to its interpretation as certain operators, giving rise to the concept of \documentclass[12pt]{minimal}\begin{document}${{\mathcal O}}$\end{document}O-operators. The \documentclass[12pt]{minimal}\begin{document}${\mathcal O}$\end{document}O-operators were in turn interpreted as tensor solutions of CYBE by enlarging the Lie algebra [Bai, C., “A unified algebraic approach to the classical Yang-Baxter equation,” J. Phys. A: Math. Theor. 40, 11073 (2007)]10.1088/1751-8113/40/36/007. The purpose of this paper is to extend this study to a more general class of operators that were recently introduced [Bai, C., Guo, L., and Ni, X., “Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras,” Commun. Math. Phys. 297, 553 (2010)]10.1007/s00220-010-0998-7 in the study of Lax pairs in integrable systems. Relations between \documentclass[12pt]{minimal}\begin{document}${\mathcal O}$\end{document}O-operators, relative differential operators, and Rota-Baxter operators are also discussed.

Formal Calculus and Umbral Calculus

We use the viewpoint of the formal calculus underlying vertex operator algebra theory to study certain aspects of the classical umbral calculus. We begin by calculating the exponential generating function of the higher derivatives of a composite function, following a very short proof which naturally arose as a motivating computation related to a certain crucial "associativity" property of an important class of vertex operator algebras. Very similar (somewhat forgotten) proofs had appeared by the 19-th century, of course without any motivation related to vertex operator algebras. Using this formula, we derive certain results, including especially the calculation of certain adjoint operators, of the classical umbral calculus. This is, roughly speaking, a reversal of the logical development of some standard treatments, which have obtained formulas for the higher derivatives of a composite function, most notably Faà di Bruno's formula, as a consequence of umbral calculus. We also show a connection between the Virasoro algebra and the classical umbral shifts. This leads naturally to a more general class of operators, which we introduce, and which include the classical umbral shifts as a special case. We prove a few basic facts about these operators.

Determinant computations for some classes of Toeplitz-Hankel matrices

The purpose of this paper is to compute the asymptotics of determinants of finite sections of operators that are trace class perturbations of Toeplitz operators. For example, we consider the asymptotics in the case where the matrices are of the form (ai−j ± ai+j+1−k)i,j=0...N−1 with k is fixed. We will show that this example as well as some general classes of operators have expansions that are similar to those that appear in the Strong Szego Limit Theorem. We also obtain exact identitities for some of the determinants that are analogous to the one derived independently by Geronimo and Case and by Borodin and Okounkov for finite Toeplitz matrices. These problems were motivated by considering certain statistical quantities that appear in random matrix theory.

Convergence and Numerics of a Multisection Method for Scattering by Three-Dimensional Rough Surfaces

We introduce a novel multisection method for the solution of integral equations on unbounded domains. The method is applied to the rough surface scattering problem in three dimensions, in particular to a Brakhage-Werner-type integral equation for acoustic scattering by an unbounded rough surface with Dirichlet boundary condition, where the fundamental solution is replaced by some appropriate half-space Green's function. The basic idea of the multisection method is to solve an integral equation $A\varphi = f$ by approximately solving the equation $P_{\varrho}A P_{\tau}\varphi = P_{\varrho} f$ for some positive constants $\varrho, \tau$. Here $P_{\varrho}$ is a projection operator that truncates a function to a ball with radius $\varrho>0$. For a very general class of operators $A$, for which the Brakhage-Werner equation from acoustic scattering is a particular example, we will show existence of approximate solutions to the multisection equation and show that approximate solutions to the multisection equation approximate the true solution $\varphi_0$ of the operator equation $A\varphi = f$. Finally, we describe a numerical implementation of the multisection algorithm and provide numerical examples for the case of rough surface scattering in three dimensions.