Showing papers on "Operator theory published in 2023"
TL;DR: In this article , it is shown that the Fueter-Sce-Qian extension operator from slice hyperholomorphic functions to monogenic functions admits various possible factorizations that induce different function spaces.
Abstract: Abstract Holomorphic functions play a crucial role in operator theory and the Cauchy formula is a very important tool to define the functions of operators. The Fueter–Sce–Qian extension theorem is a two-step procedure to extend holomorphic functions to the hyperholomorphic setting. The first step gives the class of slice hyperholomorphic functions; their Cauchy formula allows to define the so-called S -functional calculus for noncommuting operators based on the S -spectrum. In the second step this extension procedure generates monogenic functions; the related monogenic functional calculus, based on the monogenic spectrum, contains the Weyl functional calculus as a particular case. In this paper we show that the extension operator from slice hyperholomorphic functions to monogenic functions admits various possible factorizations that induce different function spaces. The integral representations in such spaces allow to define the associated functional calculi based on the S -spectrum. The function spaces and the associated functional calculi define the so-called fine structure of the spectral theories on the S-spectrum . Among the possible fine structures there are the harmonic and polyharmonic functions and the associated harmonic and polyharmonic functional calculi. The study of the fine structures depends on the dimension considered and in this paper we study in detail the case of dimension five, and we describe all of them. The five-dimensional case is of crucial importance because it allows to determine almost all the function spaces will also appear in dimension greater than five, but with different orders.
11 citations
TL;DR: In this paper , a theory for sum-by-parts (SBP) operators based on general function spaces is developed, and most of the established results for polynomial-based SBP operators carry over to this general class of operators.
Abstract: Summation-by-parts (SBP) operators are popular building blocks for systematically developing stable and high-order accurate numerical methods for time-dependent differential equations. The main idea behind existing SBP operators is that the solution is assumed to be well approximated by polynomials up to a certain degree, and the SBP operator should therefore be exact for them. However, polynomials might not provide the best approximation for some problems, and other approximation spaces may be more appropriate. In this paper, a theory for SBP operators based on general function spaces is developed. We demonstrate that most of the established results for polynomial-based SBP operators carry over to this general class of SBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently known. We exemplify the general theory by considering trigonometric, exponential, and radial basis functions.
10 citations
TL;DR: In this paper , the authors define quantum permutation matrices as matrices whose entries are operators on Hilbert spaces, and give an overview of their use in several branches of mathematics, such as quantum groups, quantum information theory, graph theory and free probability theory.
Abstract: Abstract Quantum permutations arise in many aspects of modern “quantum mathematics”. However, the aim of this article is to detach these objects from their context and to give a friendly introduction purely within operator theory. We define quantum permutation matrices as matrices whose entries are operators on Hilbert spaces; they obey certain assumptions generalizing classical permutation matrices. We give a number of examples and we list many open problems. We then put them back in their original context and give an overview of their use in several branches of mathematics, such as quantum groups, quantum information theory, graph theory and free probability theory.
7 citations
TL;DR: In this article , it was shown that integro-differential operators with non-singular kernels can be defined on larger function spaces than operators with singular kernels, as differentiability conditions can be removed.
Abstract: Integro-differential operators with non-singular kernels have been much discussed among fractional calculus researchers. We present a mathematical study to clearly establish the rigorous foundations of this topic. By considering function spaces and mapping results, we show that operators with non-singular kernels can be defined on larger function spaces than operators with singular kernels, as differentiability conditions can be removed. We also discover an analogue of the Sonine invertibility condition, giving two-sided inversion relations between operators with non-singular kernels that are not possible for operators with singular kernels.
7 citations
TL;DR: In this article , the authors considered the problem of learning the Koopman operator for discrete-time autonomous systems and showed that a representer theorem holds for the learning problem under certain but general conditions.
Abstract: In this work, we consider the problem of learning the Koopman operator for discrete-time autonomous systems. The learning problem is generally formulated as a constrained regularized empirical loss minimization in the infinite-dimensional space of linear operators. We show that a representer theorem holds for the learning problem under certain but general conditions. This allows convex reformulation of the problem in a finite-dimensional space without any approximation and loss of precision. We discuss the inclusion of various forms of regularization and constraints in the learning problem, such as the operator norm, the Frobenius norm, the operator rank, the nuclear norm, and the stability. Subsequently, we derive the corresponding equivalent finite-dimensional problem. Furthermore, we demonstrate the connection between the proposed formulation and the extended dynamic mode decomposition. We present several numerical examples to illustrate the theoretical results and verify the performance of regularized learning of the Koopman operators.
6 citations
TL;DR: In this paper , the authors used microlocal radial estimates to prove the full limiting absorption principle for P, a self-adjoint 0th order pseudodifferential operator satisfying hyperbolic dynamical assumptions as of Colin de Verdière and Saint-Raymond.
Abstract: We use microlocal radial estimates to prove the full limiting absorption principle for P, a self-adjoint 0th order pseudodifferential operator satisfying hyperbolic dynamical assumptions as of Colin de Verdière and Saint-Raymond. We define the scattering matrix for P and show that the scattering matrix extends to a unitary operator on appropriate L 2 spaces. After conjugation with natural reference operators, the scattering matrix becomes a 0th order Fourier integral operator with a canonical relation associated to the bicharacteristics of P. The operator P provides a microlocal model of internal waves in stratified fluids as illustrated in the paper of Colin de Verdière and Saint-Raymond.
6 citations
TL;DR: In this article , the authors introduced and investigated a new seminorm of operator tuples on a complex Hilbert space H when an additional semi-inner product structure defined by a positive (semi-definite) operator A on H is considered.
Abstract: The aim of this paper was to introduce and investigate a new seminorm of operator tuples on a complex Hilbert space H when an additional semi-inner product structure defined by a positive (semi-definite) operator A on H is considered. We prove the equality between this new seminorm and the well-known A-joint seminorm in the case of A-doubly-commuting tuples of A-hyponormal operators. This study is an extension of a well-known result in [Results Math 75, 93(2020)] and allows us to show that the following equalities rA(T)=ωA(T)=∥T∥A hold for every A-doubly-commuting d-tuple of A-hyponormal operators T=(T1,…,Td). Here, rA(T),∥T∥A, and ωA(T) denote the A-joint spectral radius, the A-joint operator seminorm, and the A-joint numerical radius of T, respectively.
6 citations
4 citations
TL;DR: In this paper , the authors prove the spaceability of the set of hypercyclic vectors for shift-like operators and prove weakly mixing dissipative composition operators of bounded distortion.
4 citations
21 Mar 2023
TL;DR: In this paper , the authors revisited the hypothesis needed to define the paracomposition operator, an analogue to the classic pull-back operation in the low regularity setting, first introduced by Alinhac (Commun Part Differ Equ 11(1):87-121, 1986).
Abstract: In this paper we revisit the hypothesis needed to define the “paracomposition” operator, an analogue to the classic pull-back operation in the low regularity setting, first introduced by Alinhac (Commun Part Differ Equ 11(1):87–121, 1986). More precisely we do so in two directions. First we drop the diffeomorphism hypothesis. Secondly we give estimates in global Sobolev and Zygmund spaces. Thus we fully generalize Bony’s classic paralinearasition theorem giving sharp estimates for composition in Sobolev and Zygmund spaces. In order to prove that the new class of operations benefits of symbolic calculus properties when composed by a paradifferential operator, we discuss the pull-back of pseudodifferential and paradifferential operators which then become Fourier Integral Operators. In this discussion we show that those Fourier Integral Operators obtained by pull-back are pseudodifferential or paradifferential operators if and only if they are pulled-back by a diffeomorphism that is a change of variable. We give a proof of the change of variables in paradifferential operators. Finally we study the cutoff defining paradifferential operators and it’s stability by successive composition. It is known that the cutoff becomes worse after each composition, we give a slightly refined version of the cutoffs proposed by Hörmander (Lectures on nonlinear hyperbolic differential equations, Springer, Berlin, 1997) for which give an optimal estimate on the support of the cutoff after composition.
TL;DR: In this paper , the authors extended the study of Weyl's spectrum to some classes of unbounded operators such as hypanormal, posinormal and class-operators.
Abstract: The significance of Weyl's spectrum is regional to the perturbation theory but recently it is related to operator theory in some theorems. The study of Weyl’s theorem has so far limited to the class of bounded operators, however, lately this study has been extended to some classes of unbounded operators such as hypanormal, posinormal and class- operators. The aim of this article is to continue study more spectral properties for class- operators. First, we use property to study property (b) and then show the equivalent of property (gb) with property (gw). Secondly, we start to study new spectral properties that was defined for bounded operators we start with and properties and show that for any operator that belongs to class- will possess them. Finally, we resuming our goal and give our last two properties and and prove them for , also we established the connection between these properties.
TL;DR: In this article , the centralizers and double centralizers of locally algebraic linear operators are investigated using finite topologies defined on the algebra of linear operators and the conditions under which the equality CC(A)=C(A) is fulfilled.
TL;DR: In this article , the exact order of approximation of perturbed Bernstein type operators is derived for functions analytic in a disk centered at the origin with radius greater than 1. Quantitative upper estimates for simultaneous approximation, a qualitative Voronovskaja type result and the exact ordering of approximation by these operators are obtained.
Abstract: The present paper deals with complex form of a generalization of perturbed Bernstein-type operators. Quantitative upper estimates for simultaneous approximation, a qualitative Voronovskaja type result and the exact order of approximation by these operators attached to functions analytic in a disk centered at the origin with radius greater than 1 are obtained in this study.
TL;DR: In this paper, a new construction of the Bernstein-Stancu operator is proposed which preserves the constant and e?2x, x > 0.1x approximation properties.
Abstract: Bernstein-Stancu operators are one of the most powerful tool that can be used
in approximation theory. In this manuscript, we propose a new construction
of Bernstein-Stancu operators which preserve the constant and e?2x, x > 0.
In this direction, the approximation properties of this newly defined
operators have been examined in the sense of different function spaces. In
addition to these, we present the Voronovskaya type theorem for this
operators. At the end, we provide two computational examples to demonstrate
that the new operator is an approximation procedure.
TL;DR: In this article , the (α, β)-norm of a bounded linear operator is derived for reproducing kernel Hilbert spaces, and a new upper bound involving both the Berezin number and the (β, ε)-norm is derived.
Abstract: The aim of this article was to provide improved estimates for the (α,β)-norm of a bounded linear operator. In particular, our results enabled the determination of new upper bounds involving both the Berezin number and the Berezin norm of bounded linear operators that act on reproducing kernel Hilbert spaces. Through our analysis, we hoped to enhance the understanding of the properties and behavior of such operators and contribute to the development of new mathematical tools for their characterization and application.
TL;DR: In this article , a Pólya distribution-based generalization of -Bernstein operators is proposed, which is shown to be able to interpolate at the interval's end points.
Abstract: Abstract In this manuscript, we propose a Pólya distribution-based generalization of -Bernstein operators. We establish some fundamental results for convergence as well as order of approximation of the proposed operators. We present theoretical result and graph to demonstrate the proposed operator’s intriguing ability to interpolate at the interval’s end points. In order to illustrate the convergence of proposed operators as well as the effect of changing the parameter “ ” we provide a variety of results and graphs as our paper’s conclusion.
TL;DR: It is established that uniform convergence of the operators for every function ininline-formula is established, and a quantitative result is proved.
Abstract:
In the present paper, we consider a general class of operators enriched with some properties in order to act on
TL;DR: In this paper , a class of positive linear operators based on some operators defined by Z. Walczak depend on Szãsz type operators and the famous Beta function, and the rate of convergence of a new SzÃsz-Beta type operator by using the modulus of continuity is given.
Abstract: Our purpose of this paper is to define a class of positive linear operators based on some operators defined by Z. Walczak depend on Szãsz type operators and the famous Beta function. We established a Korovkin theorem and Voronovikaja theorem. Finally, we gave the rate of convergence of a new Szãsz-Beta type operators by using the modulus of continuity.
25 Jan 2023
TL;DR: The Calderón operator is the sum of the Hardy averaging operator and its adjoint, and plays an important role in the theory of real interpolation as discussed by the authors , while the Hilbert operator arises from the continuous version of Hilbert's inequality.
Abstract: The Calderón operator is the sum of the Hardy averaging operator and its adjoint, and plays an important role in the theory of real interpolation. On the other hand, the Hilbert operator arises from the continuous version of Hilbert’s inequality. Both operators appear in different contexts and have numerous applications within harmonic analysis. In this chapter we will briefly review the Calderón and Hilbert operators, showing some of the most relevant results within functional analysis and finally we will present recent results on these operators within Fourier analysis.
14 Mar 2023
TL;DR: In this paper , Atangana-Baleanu fractional differential operators (AB-fractional differential operator) are formulated for a class of normalized analytic functions in the open unit disk.
Abstract: The majority of research on fractional differential operators focuses on functions of real variables. Atangana-Baleanu fractional differential operators (AB-fractional differential operators) are formulated in this study for a class of normalized analytic functions in the open unit disk. The recommended operators are looked at using geometric function theory principles.
TL;DR: Spectral inclusion and Hausdorff approximation of spectra and pseudospectra of generalized Schrödinger operators via approximation of subwords.
Abstract: Abstract We prove criteria, purely based on finite subwords of the potential, for spectral inclusion as well as Hausdorff approximation of pseudospectra or even spectra of generalized Schrödinger operators on the discrete line or half-line. In fact, our results are neither limited to Schrödinger or self-adjoint operators, nor to Hilbert space or 1D: By employing localized lower norms, we strongly generalize known results from the self-adjoint case to much more general and non-normal situations, including various configurations of Hamiltonians and further non-self-adjoint models with aperiodic or pseudoergodic potentials, even models with multiple varying diagonals and entries in a Banach space.
TL;DR: In this paper , a modification of Sz?sz-Mirakyan operators with a new technique that preserved the exponential functions was presented, and the convergence of these operators and modified Baskakov operators to certain functions was compared by illustrative graphics using the Mathematica algorithms.
Abstract: In this paper, we construct a modification of Sz?sz-Mirakyan operators with
a new technique that preserved the exponential functions i.e. exp(?t) and
exp(2?t), for a fixed real parameter ? > 0. We study the asymptotic
behaviour and weighted approximation of these operators. Comparisons about
one approximate better between the recent operators and the classical
Sz?sz-Mirakyan operators have also been presented. In the end, we compare
the convergence of these operators and modified Baskakov operators to
certain functions by illustrative graphics using the Mathematica algorithms.
13 Jun 2023
TL;DR: In this article , the authors used the Alughte transform and the generalized Alughtte transform to develop the Berezin radius inequality for Hilbert space operators, which is a special case of the BIR inequality.
Abstract: – In functional analysis, linear operators induced by functions are frequently encountered; thesecontain Hankel operators, constitution operators, and Toeplitz operators. The symbol of the resultantoperator is another name for the inciting function. In many instances, a linear operator on a Hilbert spaceℋ results in a function on a subset of a topological space. As a result, we regularly investigate operatorsinduced by functions, and we may also investigate functions induced by operators. The Berezin sign is awonderful representation of an operator-function relationship. F. Berezin proposed the Berezin switch in[8], and it has proven to be a vital tool in operator theory given that it utilizes many essential aspects ofsignificant operators. Many mathematicians and physicists are fascinated by the Berezin symbol of anoperator defined on the functional Hilbert space. The Berezin radius inequality has been extensively studiedin this situation by a number of mathematicians. In this paper, we use the Alughte transform and thegeneralized Alughte transform to develop Berezin radius inequalities for Hilbert space operators. Weadditionally offer fresh Berezin radius inequality results. Huban et al. [15] and Başaran et al. [6] supply theBerezin radius inequality.
TL;DR: Modifications of Green-hyperbolic operators by adding a nonlocal operator acting in a compact subset of spacetime lead to nonlocal Green operators that depend holomorphically on the parameter and have properties related to the formal dual operator.
Abstract: Green-hyperbolic operators - partial differential operators on globally hyperbolic spacetimes that (together with their formal duals) possess advanced and retarded Green operators - play an important role in many areas of mathematical physics. Here, we study modifications of Green-hyperbolic operators by the addition of a possibly nonlocal operator acting within a compact subset $K$ of spacetime, and seek corresponding '$K$-nonlocal' generalised Green operators. Assuming the modification depends holomorphically on a parameter, conditions are given under which $K$-nonlocal Green operators exist for all parameter values, with the possible exception of a discrete set. The exceptional points occur precisely where the modified operator admits nontrivial smooth homogeneous solutions that have past- or future-compact support. Fredholm theory is used to relate the dimensions of these spaces to those corresponding to the formal dual operator, switching the roles of future and past. The $K$-nonlocal Green operators are shown to depend holomorphically on the parameter in the topology of bounded convergence on maps between suitable Sobolev spaces, or between suitable spaces of smooth functions. An application to the LU factorisation of systems of equations is described.