Showing papers on "Operator theory published in 2022"
TL;DR: A universal approximation theorem of continuous multiple-input operators is proved and a novel neural operator, MIONet, is proposed, which can learn solution operators involving systems governed by ordinary and partial differential equations.
Abstract: As an emerging paradigm in scientific machine learning, neural operators aim to learn operators, via neural networks, that map between infinite-dimensional function spaces. Several neural operators have been recently developed. However, all the existing neural operators are only designed to learn operators defined on a single Banach space, i.e., the input of the operator is a single function. Here, for the first time, we study the operator regression via neural networks for multiple-input operators defined on the product of Banach spaces. We first prove a universal approximation theorem of continuous multiple-input operators. We also provide detailed theoretical analysis including the approximation error, which provides a guidance of the design of the network architecture. Based on our theory and a low-rank approximation, we propose a novel neural operator, MIONet, to learn multiple-input operators. MIONet consists of several branch nets for encoding the input functions and a trunk net for encoding the domain of the output function. We demonstrate that MIONet can learn solution operators involving systems governed by ordinary and partial differential equations. In our computational examples, we also show that we can endow MIONet with prior knowledge of the underlying system, such as linearity and periodicity, to further improve the accuracy.
150 citations
TL;DR: In this article , the authors consider the problem of learning solution operators from both linear and nonlinear advection-diffusion equations with or without reaction and find that the approximation rates depend on the architecture of branch networks as well as the smoothness of inputs and outputs of solution operators.
66 citations
TL;DR: In this paper , the authors construct neural network interpolation operators with some newly defined activation functions, and give the approximation rate by the operators for continuous functions for which they apply the K-functional and Berens-Lorentz lemma in approximation theory.
28 citations
25 citations
TL;DR: In this article, the authors investigated orthogonally additive (nonlinear) operators on C-complete vector lattices and showed that these operators preserve continuity, narrowness, compactness, and disjointness.
Abstract: In this article, we investigate orthogonally additive (nonlinear) operators on C-complete vector lattices which strongly includes all Dedekind complete vector lattices. In the first part of the paper, we present basic examples of orthogonally additive operators on function spaces. Then we show that an orthogonally additive map defined on a lateral band of a C-complete vector lattice and taking values in a Dedekind complete vector lattice could be extended to the whole space and an extended orthogonally additive operator preserves continuity, narrowness, compactness and disjointness. In the second part of the article, we consider lateral projection operators onto lateral bands. One of our main results asserts that for a C-complete vector lattice E there is a lateral projection onto every lateral band of E. Applying the technique of lateral projections we prove that for a orthogonally additive narrow operator $$T:E\rightarrow F$$
from a C-complete vector lattice E to an order continuous Banach lattice F all elements of the order interval [0, T] are narrow operators as well. Finally we show that $$T+S$$
is a narrow operator provided that the operator T is horizontally-to-norm continuous and C-compact and the operator S is narrow.
15 citations
TL;DR: In this article , a survey about recent work on weighted Banach spaces of analytic functions on the unit disc and on the whole complex plane defined with sup-norms and operators between them is presented.
Abstract: Abstract In this survey we report about recent work on weighted Banach spaces of analytic functions on the unit disc and on the whole complex plane defined with sup-norms and operators between them. Results about the solid hull and core of these spaces and distance formulas are reviewed. Differentiation and integration operators, Cesàro and Volterra operators, weighted composition and superposition operators and Toeplitz operators on these spaces are analyzed. Boundedness, compactness, the spectrum, hypercyclicity and (uniform) mean ergodicity of these operators are considered.
14 citations
TL;DR: In this paper , a new concept of functional-differential operators with constant delay on geometrical graphs that involves global delay parameter is proposed. But this concept is restricted to star-type graphs.
Abstract: We suggest a new concept of functional-differential operators with constant delay on geometrical graphs that involves global delay parameter. Differential operators on graphs model various processes in many areas of science and technology. Although a vast majority of studies in this direction concern purely differential operators on graphs (often referred to as quantum graphs), recently there also appeared some considerations of nonlocal operators on star-type graphs. In particular, there belong functional-differential operators with constant delays but in a locally nonlocal version. The latter means that each edge of the graph has its own delay parameter, which does not affect any other edge. In this paper, we introduce globally nonlocal operators that are expected to be more natural for modelling nonlocal processes on graphs. We also extend this idea to arbitrary trees, which opens a wide area for further research. Another goal of the paper is to study inverse spectral problems for operators with global delay in one illustrative case by addressing a wide range of questions including uniqueness, characterization of the spectral data as well as the uniform stability.
13 citations
9 citations
9 citations
TL;DR: The review covers the relevant theoretical elements, particularly the canonical decomposition theorem, a formulation of the learning problem, some methods to solve it, and algorithms for finding computationally efficient representations.
9 citations
TL;DR: In this article , the authors studied fundamental properties of the special affine Fourier transform (SAFT) in connection with the Fourier analysis and time-frequency analysis, and proved that if a bounded linear operator between new modulation spaces commutes with A-translation, then it is a A-convolution operator.
Abstract: We study some fundamental properties of the special affine Fourier transform (SAFT) in connection with the Fourier analysis and time-frequency analysis. We introduce the modulation space $${{\varvec{M}}}^{r,s}_A$$ in connection with SAFT and prove that if a bounded linear operator between new modulation spaces commutes with A-translation, then it is a A-convolution operator. We also establish Hörmander multiplier theorem and Littlewood-Paley theorem associated with the SAFT.
TL;DR: In this article , a new family of Szász-Mirakyan operators that depend on a non-negative parameter, say α, is defined and the convergence properties of the new operators with the aid of the Popoviciu-Bohman-Korovkin theorem-type property are presented.
Abstract: The main purpose of this paper is to define a new family of Szász–Mirakyan operators that depends on a non-negative parameter, say α. This new family of Szász–Mirakyan operators is crucial in that it includes both the existing Szász–Mirakyan operator and allows the construction of new operators for different values of α. Then, the convergence properties of the new operators with the aid of the Popoviciu–Bohman–Korovkin theorem-type property are presented. The Voronovskaja-type theorem and rate of convergence are provided in a detailed proof. Furthermore, with the help of the classical modulus of continuity, we deduce an upper bound for the error of the new operator. In addition to these, in order to show that the convex or monotonic functions produced convex or monotonic operators, we obtain shape-preserving properties of the new family of Szász–Mirakyan operators. The symmetry of the properties of the classical Szász–Mirakyan operator and of the properties of the new sequence is investigated. Moreover, we compare this operator with its classical correspondence to show that the new one has superior properties. Finally, some numerical illustrative examples are presented to strengthen our theoretical results.
TL;DR: In this article , a method for classifying and constructing the $d$-dimensional Lorentz invariant evanescent operators, which start to appear at mass dimension ten, is presented.
Abstract: Evanescent operators are a special class of operators that vanish classically in four-dimensional spacetime, while in general dimensions they are non-zero and are expected to have non-trivial physical effects at the quantum loop level in dimensional regularization. In this paper we initiate the study of evanescent operators in pure Yang-Mills theory. We develop a systematic method for classifying and constructing the $d$-dimensional Lorentz invariant evanescent operators, which start to appear at mass dimension ten. We also compute one-loop form factors for the dimension-ten operators via the $d$-dimensional unitarity method and obtain their one-loop anomalous dimensions. These operators are necessary ingredients in the study of high dimensional operators in effective field theories involving a Yang-Mills sector.
7 Mar 2022
TL;DR: This paper reviews numerical methods for approximating the Perron-Frobenius and Koopman operators, including Ulam's method and EDMD, and compares their approaches for computing finite-dimensional approximations of infinite-dimensional operators in dynamical systems.
Abstract: Information about the behavior of dynamical systems can often be obtained by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with a dynamical system. Examples of such operators are the Perron-Frobenius and the Koopman operator. In this paper, we will review different methods that have been developed over the last decades to compute finite-dimensional approximations of these infinite-dimensional operators - e.g. Ulam's method and Extended Dynamic Mode Decomposition (EDMD) - and highlight the similarities and differences between these approaches. The results will be illustrated using simple stochastic differential equations and molecular dynamics examples.
TL;DR: In this article , a new class of operators based on new type Bézier bases with a shape parameter λ and positive parameter s was constructed. But their approximation properties are illustrated on graphs for variables s, α, λ , and n .
Abstract: Abstract In the present paper, we construct a new class of operators based on new type Bézier bases with a shape parameter λ and positive parameter s . Our operators include some well-known operators, such as classical Bernstein, α -Bernstein, generalized blending type α -Bernstein and λ -Bernstein operators as special case. In this paper, we prove some approximation theorems for these operators. Approximation properties of our operators are illustrated on graphs for variables s , α , λ , and n . It should be mentioned that our operators for $\lambda =1$ λ = 1 have better approximation than Bernstein and α -Bernstein operators.
TL;DR: In this article , the authors considered pairs of weighted shift operators whose commutators are diagonal operators, when considered as operators over a general Fock space, and established a calculus for the algebra of these commutator and applied it to the general case of Gelfond-Leontiev derivatives.
Abstract: Given a weighted ℓ2 space with weights associated with an entire function, we consider pairs of weighted shift operators, whose commutators are diagonal operators, when considered as operators over a general Fock space. We establish a calculus for the algebra of these commutators and apply it to the general case of Gelfond–Leontiev derivatives. This general class of operators includes many known examples, such as classic fractional derivatives and Dunkl operators. This allows us to establish a general framework, which goes beyond the classic Weyl–Heisenberg algebra. Concrete examples for its application are provided.
TL;DR: In this paper , the maximal joint numerical range of a tuple of doubly commuting matrices has been shown to hold for Toeplitz operators as well as for tuples of normal operators.
1 Jan 2022
TL;DR: In this paper, a new concept of Q-function on partial metric spaces was introduced and a new definition of $$(\alpha,\phi,q)$$ -contractive mapping was given by considering the new kind of Q function.
Abstract: In this paper, we first introduce a new concept of Q-function on partial metric spaces. Also, we give a new definition of $$(\alpha ,\phi ,q)$$
-contractive mapping by considering the new kind of Q-function. Then we obtain some best proximity point results for such mappings. Thus, we improve and unify many well-known results in the literature. Moreover, we provide some illustrative and nontrivial examples. Therefore, we show that the approach of Haghi et al. (Topol Appl 160:450–454, 2013) cannot be applied to our results. Finally, we obtain a solution of nonlinear fractional differential equations using the new Q-function.
18 Feb 2022
TL;DR: In this article , the authors show that the square of a posinormal operator is not necessarily posinomorphic, and that powers of quasiposinormal and semi-Fredholm operators are posinomorphisms.
Abstract: Square of a posinormal operator is not necessarily posinormal$.$ But (i) powers of quasiposinormal operators are quasiposinormal and, under closed ranges assumption, powers of (ii) posinormal operators are posinormal, (iii) of operators that are both posinormal and coposinormal are posinormal and coposinormal, and (iv) of semi-Fredholm posinormal operators are posinormal.
TL;DR: In this paper , Cordero et al. introduce quasi-Banach algebras of symbol classes for Fourier integral operators that they call generalized metaplectic operators, including pseudodifferential operators.
Abstract: We generalize the results for Banach algebras of pseudodifferential operators obtained by Gröchenig and Rzeszotnik (Ann Inst Fourier 58:2279–2314, 2008) to quasi-algebras of Fourier integral operators. Namely, we introduce quasi-Banach algebras of symbol classes for Fourier integral operators that we call generalized metaplectic operators, including pseudodifferential operators. This terminology stems from the pioneering work on Wiener algebras of Fourier integral operators (Cordero et al. in J Math Pures Appl 99:219–233, 2013), which we generalize to our framework. This theory finds applications in the study of evolution equations such as the Cauchy problem for the Schrödinger equation with bounded perturbations, cf. (Cordero, Giacchi and Rodino in Wigner analysis of operators. Part II: Schrödinger equations, arXiv:2208.00505 ).
TL;DR: In this article , the authors define the two-wavelet localization operators in the setting of the Weinstein theory and give a host of sufficient conditions for the boundedness and compactness of the two wavelet localization operator.
Abstract: Daubechies first used localization operators as a mathematical tool to localized a signal in the time frequency plane. They have been a subject of research in many domains ever since. In this paper, we introduce the notion of Weinstein two-wavelet and we define the two-wavelet localization operators in the setting of the Weinstein theory. Then, we give a host of sufficient conditions for the boundedness and compactness of the two-wavelet localization operator on \(L^{p}_{\alpha }(\mathbb {R}^{d+1}_+)\) for all \(1\le p\le \infty \), in terms of properties of the symbol \(\sigma \) and the functions \(\varphi \) and \(\psi \). In the end, we study some typical examples of the Weinstein two-wavelet localization operators.
TL;DR: In this paper, it was shown that any limit law associated with a triangular array of uniformly infinitesimal random variables is infinitely divisible, and an analogous result for the convolution of measures defined on the positive half-line.
Abstract: Hincin proved that any limit law associated with a triangular array of uniformly infinitesimal random variables is infinitely divisible. Analogous results for the additive and multiplicative free convolution were proved by Bercovici, Belinschi and Pata. We prove an analogous result for the $$\boxplus _{RD}$$
convolution of measures defined on the positive half-line. This is the convolution arising from the addition of $$*$$
-free R-diagonal elements of a tracial, noncommutative probability space.
TL;DR: In this article , the authors considered Schrödinger operators with periodic potentials on periodic discrete graphs and derived trace formulas for the heat kernel and the resolvent of these operators.
TL;DR: In this article , a stochastic approach for the approximation of nonlinear Lipschitz operators in normed spaces by their eigenvectors is shown, depending on the properties of the operators themselves whether they are locally constant, (almost) linear, or convex.
Abstract: A new stochastic approach for the approximation of (nonlinear) Lipschitz operators in normed spaces by their eigenvectors is shown. Different ways of providing integral representations for these approximations are proposed, depending on the properties of the operators themselves whether they are locally constant, (almost) linear, or convex. We use the recently introduced notion of eigenmeasure and focus attention on procedures for extending a function for which the eigenvectors are known, to the whole space. We provide information on natural error bounds, thus giving some tools to measure to what extent the map can be considered diagonal with few errors. In particular, we show an approximate spectral theorem for Lipschitz operators that verify certain convexity properties.