Composition Operators Research Articles

Examining the behavior of parametric convex operators on a certain set of analytical functions

Mathematical operators that maintain convex functional combinations involving at least one parameter are called parametric convex operators (PCOs) on analytic function spaces. In the context of analytic functions, these operators are often described on spaces of holomorphic (analytic) functions. The most important challenge in this direction is to investigate the boundedness and discovers the upper bound element (in this case, it is the extreme analytic function in the open unit disk). The boundedness of weighted composition operators has been studied extensively on several analytic function spaces. In the present study, we examine the boundedness of two proposed weighted composition operators on different analytic function spaces with differences structure. Examples are illustrated containing the Koebe function (the extreme convex function in the open unit disk).The methodology of this study based on:•Define a new convex operator with a set of parameters.•A special subclass of analytic functions in the open unit disk including the proposed convex operator is suggested.•The boundedness of the convex operator is investigated by illustrating a set of conditions on the parameters.

A new class of Carleson measures and integral operators on Bergman spaces

Let n be a positive integer and u=(u0,u1,…,un) with uk∈H(D) for 0≤k≤n, where H(D) is the space of analytic functions in the unit disk D. For 0<p,q<∞, we introduced a new class of (u,p,q)-Sobolev Carleson measure for the Bergman space Ap, i.e. a positive Borel measure μ on D, for which there exists a constant C>0 such that∫D|u0(z)f(z)+u1(z)f′(z)+⋯+un(z)f(n)(z)|qdμ(z)≤C‖f‖pq for all f∈Ap. Using Sobolev Carleson measures, we characterized the boundedness and compactness of the generalized Volterra-type operator Ig(n) acting on Bergman space to another, which is represented asIg(n)f=In(fg0+f′g1+⋯+f(n−1)gn−1), here g=(g0,⋯,gn−1) with gk∈H(D) for 0≤k≤n−1 and (If)(z)=∫0zf(w)dw is the usual integration operator. This operator is a generalization of the operator introduced by Chalmoukis in [5]. As a consequence, we obtain conditions for certain linear differential equations to have solutions in Bergman spaces. Moreover, we study the boundedness, compactness and Hilbert-Schmidtness of the following sums of generalized weighted composition operators: Lu,φ(n)=∑k=0nWuk,φ(k), Where φ is an analytic self-map of D and Wuk,φf=uk⋅f(k)∘φ.

Generalized Counting Functions and Composition Operators on Weighted Bergman Spaces of Dirichlet Series

Generalized Counting Functions and Composition Operators on Weighted Bergman Spaces of Dirichlet Series

Compact linear combinations of composition operators on Hardy spaces

Compact linear combinations of composition operators on Hardy spaces

Testing AI on language comprehension tasks reveals insensitivity to underlying meaning.

Large Language Models (LLMs) are recruited in applications that span from clinical assistance and legal support to question answering and education. Their success in specialized tasks has led to the claim that they possess human-like linguistic capabilities related to compositional understanding and reasoning. Yet, reverse-engineering is bound by Moravec's Paradox, according to which easy skills are hard. We systematically assess 7 state-of-the-art models on a novel benchmark. Models answered a series of comprehension questions, each prompted multiple times in two settings, permitting one-word or open-length replies. Each question targets a short text featuring high-frequency linguistic constructions. To establish a baseline for achieving human-like performance, we tested 400 humans on the same prompts. Based on a dataset of n = 26,680 datapoints, we discovered that LLMs perform at chance accuracy and waver considerably in their answers. Quantitatively, the tested models are outperformed by humans, and qualitatively their answers showcase distinctly non-human errors in language understanding. We interpret this evidence as suggesting that, despite their usefulness in various tasks, current AI models fall short of understanding language in a way that matches humans, and we argue that this may be due to their lack of a compositional operator for regulating grammatical and semantic information.

Iterates of composition operators on global spaces of ultradifferentiable functions

We analyze the behaviour of the iterates of composition operators defined by polynomials acting on global classes of ultradifferentiable functions of Beurling type and being invariant under Fourier transform. We characterize the polynomials ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document} for which the sequence of iterates is equicontinuous between two different Gelfand–Shilov spaces. For the particular case in which the weight ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} is equivalent to a power of the logarithm, the result obtained characterizes the polynomials ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document} for which the composition operator Cψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_\\psi $$\\end{document} is power bounded in Sω(R).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {S}}}_\\omega ({{\\mathbb {R}}}).$$\\end{document} Unlike the composition operators in the Schwartz class, the Waelbroek spectrum of an operator Cψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_\\psi $$\\end{document}, ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document} being a polynomial of degree greater than one lacking fixed points is never compact. We focus on the problem of the convergence of Neumann series. We deduce the continuity of the resolvent operator between two different Gelfand–Shilov classes for polynomials ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document} lacking fixed points. Concerning polynomials of second degree the most interesting case is the one in which the polynomial only has one fixed point: we provide some restrictions on the indices d,d′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d, d'$$\\end{document} that are necessary for the resolvent operator to be continuous between the Gelfand–Shilov classes Σd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma _d$$\\end{document} and Σd′.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma _{d'}.$$\\end{document}

Assessing National Governance Modernization: Leveraging the Ordered Weighted Geometric Composition Operator

The national governance modernization is an important part of Chinese-style modernization. It is of great importance to advance the modernization process by assessing the modernization level of national governance. In this paper, we propose a new method for assessing the national governance modernization using the ordered weighted geometric composition operator (OWGC). First, defining the OWGC by introducing the average operator. The application of the OWGC in measuring the national governance modernization is then demonstrated and the general measurement procedure is presented. Finally, empirical data are used to analyze the degree and regional differences in national governance modernization. The empirical results obtained by using the OWGC not only show uneven development between different provinces, but also reflect the development strengths and weaknesses of each region, which will help provide guidelines for improving the level of modernization and promoting coordinated regional development delivery.

Renormalized primordial black holes

The formation of primordial black holes in the early universe may happen through the collapse of large curvature perturbations generated during a non-attractor phase of inflation or through a curvaton-like dynamics after inflation. The fact that such small-scale curvature perturbation is typically non-Gaussian leads to the renormalization of composite operators built up from the smoothed density contrast and entering in the calculation of the primordial black abundance. Such renormalization causes the phenomenon of operator mixing and the appearance of an infinite tower of local, non-local and higher-derivative operators as well as to a sizable shift in the threshold for primordial black hole formation. This hints that the calculation of the primordial black hole abundance is more involved than what generally assumed. We show the impact of this phenomenon in a perturbatively non-gaussian scenario, giving also an estimate of its effect on the threshold for primordial black hole formation.

Order-Bounded Difference in Weighted Composition Operators Between Fock Spaces

There are two aims in this paper. The first aim is to characterize the order-bounded weighted composition operators between Fock spaces, and the second is to further characterize the order-bounded difference in weighted composition operators between Fock spaces. At the same time, six examples are given to illustrate the relations between boundedness and ordered boundedness. Moreover, an interesting result is found that differences in weighted composition operators defined by some special weighted functions and symbol functions are order-bounded between Fock spaces if and only if each weighted composition operator is compact between Fock spaces. Finally, two open questions are also put forward for converting larger Fock spaces into smaller ones.

Remark on square roots of self-adjoint weighted composition operators on H2

In this paper, we study square roots of self-adjoint weighted composition operators on H2. More precisely, we focus on square roots Wf,φ of a self-adjoint operator Wg,ψ=Wf,φ2 when φ is a linear fractional selfmap of D. We also investigate several properties of such Wf,φ. Finally, we show that the square roots Wf,φ may be other, nonself-adjoint weighted composition operators and have nontrivial invariant subspaces.

On (p, q)-eigenvalues of the weighted p-Laplace operator in outward Hölder cuspidal domains

AbstractIn this paper, we will study Neumann (p, q)-eigenvalue problem for the weighted p-Laplace operator in outward Hölder cuspidal domains. The suggested method is based on the composition operators on weighted Sobolev spaces.

Finite Sum of Integral Operators from the Fractional Cauchy Spaces to Bloch-Type and Zygmund-Type Spaces

The family of fractional Cauchy transforms, defined on the open unit disc in the complex plane, is of classical and modern interest. Membership of an analytic function in the family is determined by the requirement that the function can be expressed as an integral of a certain kernel against a complex Borel measure on the disc. Such an integral representation imposes a growth condition on the function and its derivatives. This exposes a connection between the families of Cauchy transforms and familiar spaces of analytic functions, such as the Bloch spaces and the Zygmund space. The notion of a composition operator has been a fruitful area of study. More generally, many authors have studied weighted composition operators, the differentiation operator, integral-type operators, and various products of such operators, acting from one normed linear space of analytic functions to another such space. A common theme of such works is to characterize the operator-theoretic notions of boundedness and compactness in terms of the inducing symbols of the operator. We extend these studies to a specific linear transformation which will be defined as the sum of finitely many integral operators. Our conclusions include a complete characterization of boundedness and compactness of the integral sum, acting from the fractional Cauchy spaces to the Bloch-type and Zygmund-type spaces.

The Spectra of Multiplication Operators and Weighted Composition Operators on Iterated Weighted-Type Banach Spaces of Analytic Functions

This research aims to analyze the spectral of multiplication operators acting on weighted Banach spaces of analytic functions defined on the unit disk. These spaces, denoted by Sn : n ∈ N, include well-known cases such as the Bloch space and the Zygmund space for n = 1 and n = 2, respectively. Additionally, we offer a description of the spectra of weighted composition operators within these spaces. The outline of this paper provides a structured framework for organizing the research, starting from the introduction to the conclusion and references. The main section is to investigate the spectrum of multiplication operators on Sn spaces, and it followed by a section that is designed to build upon the previous one, leading to characterize the spectrum of weighted composition operators on the same spaces.

Linear Dynamics of Multiplication and Composition Operators on $$\textrm{Hol}(\mathbb {D})$$

Linear Dynamics of Multiplication and Composition Operators on $$\textrm{Hol}(\mathbb {D})$$

Enhancing network robustness with structural prior and evolutionary techniques

Robustness optimization in complex networks is a critical research area due to its implications for the reliability and stability of various systems. However, existing algorithms encounter two key challenges: the lack of integration of prior network knowledge, leading to suboptimal solutions, and high computational costs, which hinder their practical application. To address these challenges, this paper introduces Eff-R-Net, an efficient evolutionary algorithm framework aimed at enhancing the robustness of complex networks through accelerated evolution. Eff-R-Net leverages global and local network information, featuring a novel three-part composite crossover operator. Prior network knowledge is incorporated in mutation and local search operators to expedite the construction of networks with superior robustness. Additionally, a simplified method for calculating robustness enhances efficiency, while adaptive hyper-parameters dynamically adjust operators execution probabilities for optimal evolution. Extensive evaluations on both synthetic (Scale-Free, Erdös-Rényi, and Small-World) and three infrastructure real-world networks demonstrate the superiority of Eff-R-Net. The algorithm improves robustness by 12.8% and reduces computational time by 25.4% compared to state-of-the-art algorithm in real-world network experiments. These findings underscore Eff-R-Net's versatility and potential in enhancing network robustness across different domains.

On the Product of Weighted Composition Operators and Radial Derivative Operators from the Bloch-Type Space into the Bers-Type Space on the Fourth Loo-Keng Hua Domain

Let HEIV be the fourth Loo-Keng Hua domain. We study the boundedness of the product of the weighted composition operator and the radial derivative operator from the Bloch-type space Bα(HEIV) into the Bers-type space Aβ(HEIV) and provide the necessary and sufficient conditions for their boundedness.

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.

Approximation of operators on real line

There are exponential operators which are defined on the whole real line, namely, the Ismail–May operators , connected to , and the Gauss–Weierstrass operators , connected with 1. Here we consider the composition of these two operators and observe that the composition operator is not of exponential type operator. We find moment‐generating functions and moments of these operators and estimate some point‐wise convergence of these operators using the characteristic function. We also estimate quantitative Voronovskaya type asymptotic formula and error estimation. Also, the difference of the composition operators with its components and is estimated in terms of modulus of continuity. Finally, we provide some graphs to see visualize the convergence by considering the function .

M-Isometric Composition Operators on Discrete Spaces

In this paper we study composition operators on discrete spaces L2(μ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ L^{2}(\\mu ) $$\\end{document}. We establish the classification of underlying graphs of such operators. We obtain the characterization of m-isometric composition operators such that the underlying graph has one cycle. The paper contains also the solution of the Cauchy dual subnormality problem in the class of such composition operators.

Composition operators on the algebra of Dirichlet series

The algebra of Dirichlet series A(C+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}({{\\mathbb {C}}}_+)$$\\end{document} consists on those Dirichlet series convergent in the right half-plane C+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {C}}}_+$$\\end{document} and which are also uniformly continuous there. This algebra was recently introduced by Aron, Bayart, Gauthier, Maestre, and Nestoridis. We describe the symbols Φ:C+→C+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi :{{\\mathbb {C}}}_+\\rightarrow {{\\mathbb {C}}}_+$$\\end{document} giving rise to bounded composition operators CΦ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{\\Phi }$$\\end{document} in A(C+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}({{\\mathbb {C}}}_+)$$\\end{document} and denote this class by GA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}_{\\mathcal {A}}$$\\end{document}. We also characterise when the operator CΦ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{\\Phi }$$\\end{document} is compact in A(C+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}({{\\mathbb {C}}}_+)$$\\end{document}. As a byproduct, we show that the weak compactness is equivalent to the compactness for CΦ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{\\Phi }$$\\end{document}. Next, the closure under the local uniform convergence of several classes of symbols of composition operators in Banach spaces of Dirichlet series is discussed. We also establish a one-to-one correspondence between continuous semigroups of analytic functions {Φt}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{\\Phi _t\\}$$\\end{document} in the class GA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}_{\\mathcal {A}}$$\\end{document} and strongly continuous semigroups of composition operators {Tt}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{T_t\\}$$\\end{document}, Ttf=f∘Φt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_tf=f\\circ \\Phi _t$$\\end{document}, f∈A(C+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\in \\mathcal {A}({{\\mathbb {C}}}_+)$$\\end{document}. We conclude providing examples showing the differences between the symbols of bounded composition operators in A(C+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}({{\\mathbb {C}}}_+)$$\\end{document} and the Hardy spaces of Dirichlet series Hp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}^p$$\\end{document} and H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}^{\\infty }$$\\end{document}.