Abstract: Introduction- Fredholm Operators and Riesz Theory- Abstract Cauchy Problem- Fredholm Theory Related to Some Measures- Pertubation Results- Essential Spectra of Linear Operators- Essentia Pseudo-spectra- S-Essential Spectra- Essential Spectra of 2 X 2 Block Operator Matrices- Essential Spectra of 3 X 3 Block Operator Matrices- Applications in Mathematical Physics and Biology

Abstract: Koopman operator is a composition operator defined for a dynamical system described by nonlinear differential or difference equation. Although the original system is nonlinear and evolves on a finite-dimensional state space, the Koopman operator itself is linear but infinite-dimensional (evolves on a function space). This linear operator captures the full information of the dynamics described by the original nonlinear system. In particular, spectral properties of the Koopman operator play a crucial role in analyzing the original system. In the first part of this paper, we review the so-called Koopman operator theory for nonlinear dynamical systems, with emphasis on modal decomposition and computation that are direct to wide applications. Then, in the second part, we present a series of applications of the Koopman operator theory to power systems technology. The applications are established as data-centric methods, namely, how to use massive quantities of data obtained numerically and experimentally, through spectral analysis of the Koopman operator: coherency identification of swings in coupled synchronous generators, precursor diagnostic of instabilities in the coupled swing dynamics, and stability assessment of power systems without any use of mathematical models. Future problems of this research direction are identified in the last concluding part of this paper.

Abstract: The concern of this paper is to introduce a Kantorovich modification of $( p,q ) $ -Baskakov operators and investigate their approximation behaviors. We first define a new $( p,q ) $ -integral and construct the operators. The rate of convergence in terms of modulus of continuities, quantitative and qualitative results in weighted spaces, and finally pointwise convergence of the operators for the functions belonging to the Lipschitz class are discussed.

Abstract: In this paper the iterates of the $$q$$ -Durrmeyer operators are introduced using a modification. For these iterates the convergence results are obtained. The estimates for the rate of convergence are obtained in terms of the modulus of smoothness. A Voronovskaya type asymptotic result is obtained. Finally necessary conditions are derived which guarantee the convergence of these iterates to the projection operators.

Abstract: In recent years, it has been well understood that a Calderon-Zygmund operator $T$ is pointwise controlled by a finite number of dyadic operators of a very simple structure (called the sparse operators). We obtain a similar pointwise estimate for the commutator $[b,T]$ with a locally integrable function $b$. This result is applied into two directions. If $b\in BMO$, we improve several weighted weak type bounds for $[b,T]$. If $b$ belongs to the weighted $BMO$, we obtain a quantitative form of the two-weighted bound for $[b,T]$ due to Bloom-Holmes-Lacey-Wick.

Abstract: We incorporate gauge-invariant local composite operators into the twistor-space formulation of N=4 super Yang-Mills theory. In this formulation, the interactions of the elementary fields are reorganized into infinitely many interaction vertices and we argue that the same applies to composite operators. To test our definition of the local composite operators in twistor space, we compute several corresponding form factors, thereby also initiating the study of form factors using the position twistor-space framework. Throughout this Letter, we use the composite operator built from two identical complex scalars as a pedagogical example; we treat the general case in a follow-up paper.

Abstract: We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations. We develop a new method (F-method) based on the algebraic Fourier transform for generalized Verma modules, which characterizes differential symmetry breaking operators by means of certain systems of partial differential equations. In contrast to the setting of real flag varieties, continuous symmetry breaking operators of Hermitian symmetric spaces are proved to be differential operators in the holomorphic setting. In this case, symmetry breaking operators are characterized by differential equations of second order via the F-method.

Abstract: 1.Introduction -- 2. Dynamic string-averaging methods in Hilbert spaces -- 3. Iterative methods in metric spaces -- 4. Dynamic string-averaging methods in normed spaces -- 5. Dynamic string-maximum methods in metric spaces -- 6. Spaces with generalized distances -- 7. Abstract version of CARP algorithm -- 8. Proximal point algorithm -- 9. Dynamic string-averaging proximal point algorithm -- 10. Convex feasibility problems -- 11. Iterative subgradient projection algorithm -- 12. Dynamic string-averaging subgradient projection algorithm.– References.– Index. .

Abstract: We prove existence and conditional energetic stability of solitary-wave solutions for the two classes of pseudodifferential equations \begin{equation*} u_t+\left(f(u)\right)_x-\left(L u\right)_x=0 \end{equation*} and \begin{equation*} u_t+\left(f(u)\right)_x+\left(L u\right)_t=0, \end{equation*} where $f$ is a nonlinear term, typically of the form $c|u|^p$ or $cu|u|^{p-1}$, and $L$ is a Fourier multiplier operator of positive order. The former class includes for instance the Whitham equation with capillary effects and the generalized Korteweg-de Vries equation, and the latter the Benjamin-Bona-Mahony equation. Existence and conditional energetic stability results have earlier been established using the method of concentration-compactness for a class of operators with symbol of order $s\geq 1$. We extend these results to symbols of order $0 < s < 1$, thereby improving upon the results for general operators with symbol of order $s\geq 1$ by enlarging both the class of linear operators and nonlinearities admitting existence of solitary waves. Instead of using abstract operator theory, the new results are obtained by direct calculations involving the nonlocal operator $L$, something that gives us the bounds and estimates needed for the method of concentration-compactness.

Abstract: We introduce the notion of generalized weighted Morrey spaces and investigate the boundedness of some operators in these spaces, such as the Hardy–Littlewood maximal operator, generalized fractional maximal operators, generalized fractional integral operators, and singular integral operators. We also study their boundedness in the vector-valued setting.

Abstract: In this article, we derive several multidimensional Hilbert-type inequalities, including certain differential operators. Further, we determine the conditions under which the constants appearing on the right-hand sides of the established inequalities are the best possible. As an application, some particular examples are also studied.

Abstract: In this paper, we study an extension of the bivariate Lupas–Durrmeyer operators based on Polya distribution. For these operators we get a Voronovskaja type theorem and the order of approximation using Peetre’s K-functional. Then, we construct the Generalized Boolean Sum operators of Lupas–Durrmeyer type and estimate the degree of approximation in terms of the mixed modulus of smoothness.

Abstract: In this paper we prove a Feynman–Kac–Ito formula for magnetic Schrodinger operators on arbitrary weighted graphs. To do so, we have to provide a natural and general framework both on the operator theoretic and the probabilistic side of the equation. On the operator side we identify a very general class of potentials that allows the definition of magnetic Schrodinger operators. On the probabilistic side, we introduce an appropriate notion of stochastic line integrals with respect to magnetic potentials. Apart from linking the world of discrete magnetic operators with the probabilistic world through the Feynman–Kac–Ito formula, the insights from this paper gained on both sides should be of an independent interest. As applications of the Feynman–Kac–Ito formula, we prove a Kato inequality, a Golden–Thompson inequality and an explicit representation of the quadratic form domains corresponding to a large class of potentials.

Abstract: In this paper we focus on Hermite subdivision operators that act on vector valued data interpreting their components as function values and associated consecutive derivatives. We are mainly interested in studying the exponential and polynomial preservation capability of such kind of operators, which can be expressed in terms of a generalization of the spectral condition property in the spaces generated by polynomials and exponential functions. The main tool for our investigation are convolution operators that annihilate the aforementioned spaces, which apparently is a general concept in the study of various types of subdivision operators. Based on these annihilators, we characterize the spectral condition in terms of factorization of the subdivision operator.

Abstract: 1. Let L2 be the usual Lebesgue space with respect to Haar measure (normalized to one) on the unit circle and H2 those elements of L2 whose Fourier transforms vanish on the negative integers. The unitary map which takes any element of L2 into its Fourier transform in I2 (the space of square summable sequences on the integers) will take H2 unitarily onto /+ (the sequences of I2 whose values vanish on the negative integers). Let qb be a bounded measurable function on the circle group and P the projection from L2 onto H2. By the Laurent operator L^ on L2 into itself we shall simply mean the operator defined by L^g = qbg. The Toeplitz operator T¿ shall be PL¿ restricted to H2. The unitary Parseval operator from L2 onto I2 will induce corresponding Laurent and Toeplitz operators on I2 and l\ respectively which, of course, will be operators defined by convolution equations. The space H2 is clearly the closure in L2 of the algebra A consisting of continuous functions on the circle group whose Fourier coefficients vanish on the negative integers. The algebra A is a prototype of what in the past few years has come to be called a Dirichlet algebra. As in the circle group case, a general Dirichlet algebra A can be used to obtain L2 and H2 spaces and Laurent and Toeplitz operators can be defined. We are interested in the question under what circumstances a Toeplitz operator is invertible; i.e., is a one-one map of H2 onto itself? We have solved this problem by showing that the problem of invertibility of a general Toeplitz operator can be reduced to the problem of invertibility of a special type of Toeplitz operator and that, in turn, this latter problem is equivalent to a problem solved by Helson and Szegö [6]. The invertibility problem on the transform space of H2 of the circle group, i.e., the invertibility problem of Toeplitz operators on I2, is of course the problem of solving a discrete Wiener-Hopf type convolution equation. Moreover, by a simple transformation the solution of the problem on l\ leads to a solution of the usual H2 type of Wiener-Hopf equation on the real line. The problem of invertibility for I2., also using the ideas of [6], was solved in 1960 by H. Widom and announced by him, in part, in [13]. Without knowledge of Widom's work, we subsequently rediscovered his results and also found that the methods we had used would work

Abstract: In the present paper, considering the simulation function, we give a new class of Picard operators on

Abstract: In this paper we introduce and study a new sequence of positive linear operators acting on function spaces defined on a convex compact subset. Their construction depends on a given Markov operator, a positive real number and a sequence of probability Borel measures. By considering special cases of these parameters for particular convex compact subsets we obtain the classical Kantorovich operators defined in the one-dimensional and multidimensional setting together with several of their wide-ranging generalizations scattered in the literature. We investigate the approximation properties of these operators by also providing several estimates of the rate of convergence. Finally, the preservation of Lipschitz-continuity as well as of convexity are discussed

Abstract: In this paper we study the spectral properties of (m, C)-isometric operators. In particular, if $$T\in \mathcal{{L(H)}}$$ is (m, C)-isometric operators, then the power of (m, C)-isometric operators is also (m, C)-isometric operators. Moreover, if $$T^{*}$$ has the single-valued extension property, then T has the single-valued extension property. Finally, we investigate conditions for (m, C)-isometric operators to be (1, C)-isometric operators.

Abstract: In the present paper, we study the applications of the extension of quantum calculus based on two parameters. We define beta function and establish an identity with gamma function, for two parameters $(p,q)$, i e. the post-quantum calculus. We also propose the $(p,q)$-Durrmeyer operators, estimate moments and establish some direct results. Depending on the selection of $p$ and $q,$ the rate of convergence of the our new operators can provide better approximation than those of the Bernstein-Durrmeyer operators and its $q$-analogue. In the end, we provide some graphs using the software Mathematica.