Certain Class Of Operators Research Articles

On wide-(s) sequences and their applications to certain classes of operators

A basic sequence in a Banach space is called wide-(s) if it is bounded and dom- inates the summing basis. (Wide-(s) sequences were originally introduced by I. Singer, who termed them P � -sequences). These sequences and their quantified versions, termed λ-wide- (s) sequences, are used to characterize various classes of operators between Banach spaces, such as the weakly compact, Tauberian, and super-Tauberian operators, as well as a new in- termediate class introduced here, the strongly Tauberian operators. This is a nonlocalizable class which nevertheless forms an open semigroup and is closed under natural operations such as taking double adjoints. It is proved for example that an operator is non-weakly compact iff for every e > 0, it maps some (1+e)-wide-(s)-sequence to a wide-(s) sequence. This yields the quantitative triangular arrays result characterizing reflexivity, due to R.C. James. It is shown that an operator is non-Tauberian (resp. non-strongly Tauberian) iff for every e > 0, it maps some (1 + e)-wide-(s) sequence into a norm-convergent sequence (resp. a sequence whose image has diameter less than e). This is applied to obtain a direct characterization of super-Tauberian operators, as well as the following characterization, which strengthens a recent result of M. Gonzalez and A. Mart´inez-Abejon: An operator is non-super-Tauberian iff there are for every e > 0, finite (1+ e)-wide-(s) sequences of arbitrary length whose images have norm at most e.

Improving the order and rates of convergence for the Super-Halley method in Banach spaces

In this study we are concerned with the problem of approximating a locally unique solution of an equation on a Banach space. A semilocal convergence theorem is given for the Super-Halley method in Banach spaces. Earlier results have shown that the order of convergence is four for a certain class of operators [4], [5], [8]. These results were not given in affine invariant form, and made use of a real quadratic majorizing polynomial. Here, we provide our results in affine invariant form showing that the order of convergence is at least four. In cases that it is exactly four the rate of convergence is improved. We achieve these results by using a cubic majorizing polynomial. Some numerical examples are given to show that our error bounds are better than earlier ones.

Global Solvability on the Torus for Certain Classes of Operators in the Form of a Sum of Squares of Vector Fields

Global Solvability on the Torus for Certain Classes of Operators in the Form of a Sum of Squares of Vector Fields

On Some Properties of Generalized Proximal Point Methods for Variational Inequalities

We discuss here generalized proximal point methods applied to variational inequality problems. These methods differ from the classical point method in that a so-called Bregman distance substitutes for the Euclidean distance and forces the sequence generated by the algorithm to remain in the interior of the feasible region, assumed to be nonempty. We consider here the case in which this region is a polyhedron (which includes linear and nonlinear programming, monotone linear complementarity problems, and also certain nonlinear complementarity problems), and present two alternatives to deal with linear equality constraints. We prove that the sequences generated by any of these alternatives, which in general are different, converge to the same point, namely the solution of the problem which is closest, in the sense of the Bregman distance, to the initial iterate, for a certain class of operators. This class consists essentially of point-to-point and differentiable operators such that their Jacobian matrices are positive semidefinite (not necessarily symmetric) and their kernels are constant in the feasible region and invariant through symmetrization. For these operators, the solution set of the problem is also a polyhedron. Thus, we extend a previous similar result which covered only linear operators with symmetric and positive-semidefinite matrices.

Operator product expansions in four-dimensional superconformal field theories

The operator product expansion in four-dimensional superconformal field theory is discussed. It is demonstrated that the OPE takes a particularly simple form for certain classes of operators. These are chiral operators, principally of interest in theories with N = 1 or N = 2 supersymmetry, and analytic operators, of interest in N = 2 and N = 4. It is argued that the Green's functions of such operators can be determined up to constants.

Relatively bounded and compact perturbations of nth order differential operators

A perturbation theory for nth order differential operators is developed. For certain classes of operators L, necessary and sufficient conditions are obtained for a perturbing operator B to be relatively bounded or relatively compact with respect to L. These perturbation conditions involve explicit integral averages of the coefficients of B. The proofs involve interpolation inequalities.

On an algebra of a certain class of operators in a slab domain in $R^2$

In this paper a Dirichlet problem for the Laplacian in a domain with a corner in R2 is treated and a new method of construction of a local parametrix for that Dirichlet problem at a corner point is given. In order to construct a parametrix a certain class of operators in a slab domain is introduced and it is shown that that operator class is an algebra.

Power methods for calculating eigenvalues and eigenvectors of spectral operators on Hilbert spaces

Power, inverse power and orthogonal iteration algorithms for determining the eigenstructure of operators on infinite-dimensional Hilbert spaces are developed. Convergence properties of the algorithms are established for certain classes of operators associated with the control of systems described by partial differential equations. A simple finite-difference method is applied to a particular operator to illustrate the utility of the algorithms.

SOME FUNCTIONAL ANALYTIC PROPERTIES OF COMPOSITION OPERATORS

Abstract This is a report on a number of recent results on composition operators which map, for 0 < p ⪯ q ∞, the Hardy space Hp (on the unit disk in the complex plane) into H q. Attention is focused on questions of boundedness (existence), compactness, order boundedness and, in connection with the latter, on relating the absolutely summing and nuclearity character as well as special factorization properties of the operator to function theoretic properties of the defining symbol. Moreover, tools are provided to show that certain classes of operators can well be distinguished already on the level of composition operators.

Global Analytic Hypoellipticity of a Class of Degenerate Elliptic Operators on the Torus

In this paper we prove global analytic hypoellipticity on the torus for certain classes of operators in the form of a sum of squares of vector fields with real-valued and real analytic coefficients. The main conditions are that every point of the torus is of finite type for the vector fields and that the coefficients are independent of some of the variables. We show, for instance, that the well known not locally analytic hypoelliptic operator of Baouendi and Goulaouic [1] is globally analytic hypoelliptic on the torus. Also, we show that the operators that were proved in Hanges-Himonas [11] and in Christ [6], [7] to be not locally analytic hypoelliptic, are globally analytic hypoelliptic. Global analytic hypoellipticity of sum of squares operators on a compact analytic manifold M is an open problem except for certain cases. From the complex analysis point of view, the most important situation is when M is the boundary of a domain in C n and the operator is Kohn’s Laplacian, b, for M. In the symplectic case the global analytic hypoellipticity of b follows from the fundamental result of Tartakoff [24], and Treves [26]. In

Radii of convergence and index for 𝑝-adic differential operators

We study the radii of p p -adic convergence of solutions at a generic point of homogeneous linear differential operators whose coefficients are analytic elements. As an application we prove a conjecture of P. Robba (for a certain class of operators) concerning the relation between radii of convergence and index on analytic elements. We also give an explicit factorization theorem for p p -adic differential operators, based on the radii of generic convergence and the slopes of the associated Newton polygon.

The point spectra for certain classes of operators in B(? p )

In a recent paper [5] Maddox determined the point spectrum of the Cesaro matrices of order α>0, considered as series-to-series operators. In this paper we obtain the point spectrum for the Endl generalized Hausdorff matrices, the Hausdorff matrices, and for a wide class of generalized Hausdorff matrices, which include the Endl and ordinary Hausdorff matrices as special cases. We also obtain point spectrum results for a wide class of weighted mean matrices. Since the Cesaro matrices are examples of Hausdorff matrices, our results contain the corresponding theorems of Maddox as special cases.

Convexity properties of distinguished eigenvalues of certain classes of operators

We prove two convexity results: (1) Let A ( ε ) = A 0 + ε A 1 A(\varepsilon ) = {A_0} + \varepsilon {A_1} be a family of selfadjoint operators in a Krein space with separated spectrum so that the maximum λ _ ( ε ) \lambda \_(\varepsilon ) of the spectrum of negative type of A ( ε ) A(\varepsilon ) is an isolated simple eigenvalue. Then λ _ ( ε ) \lambda \_(\varepsilon ) is convex. (2) Let λ _ ( ε ) \lambda \_(\varepsilon ) be the left distinguished eigenvalue of the generalized eigenvalue problem ( A 0 + ε A 1 ) x = λ B x ({A_0} + \varepsilon {A_1})x = \lambda Bx where A 1 {A_1} and B B are real diagonal matrices and A 0 {A_0} is an irreducible essentially nonnegative matrix. Then λ _ ( ε ) \lambda \_(\varepsilon ) is convex. In both cases λ _ ( ε ) \lambda \_(\varepsilon ) is strictly convex unless it is linear.

On the spectra of hypoelliptic operators in R 2

Sufficient conditions are given for the spectra and essential spectra of certain classes of operators in R2 to be contained in an interval of the form [d,∞).

Uniform Closure of Certain Classes of Operators

Uniform Closure of Certain Classes of Operators

Positive p-summing operators, vector measures and tensor products

In this paper we shall introduce a certain class of operators from a Banach lattice X into a Banach space B (see Definition 1) which is closely related to p-absolutely summing operators defined by Pietsch [8].These operators, called positive p-summing, have already been considered in [9] in the case p = 1 (there they are called cone absolutely summing, c.a.s.) and in [1] by the author who found this space to be the space of boundary values of harmonic B-valued functions in .Here we shall use these spaces and the space of majorizing operators to characterize the space of bounded p-variation measures and to endow the tensor product with a norm in order to get as its completion in this norm.

Quasi-Affinity in certain Classes of Operators

The family of operators S + ζVα (ζ, α€C, Re α > 0), where V is an injective S-Volterra operator (that is, [S, V[= V2) and — A ≡ — V−1 generates a uniformly bounded C0-semigroup, is studied in the context of similarity and of the weaker quasi-affinity relation. It is shown that S is similar to S + ζVα for all ζ, α€C, Re α > 1, and is a quasi-affine transform of S + tVα for all t ⩾ 0 and 0 < α < 1.

Local ergodicity as a probe for chaos in quantum systems: Application to the Henon–Heiles system

Classical chaos is usually accompanied by ‘‘local’’ ergodicity—ergodic behavior in regions of phase space that are generally smaller than, but of the same dimensionality as, the energy surface. If there are strong bottlenecks in phase space impeding the relaxation to statistical equilibrium in the full region of ergodicity, it is often possible to define an approximate form of ergodic behavior in a smaller subregion where relaxation occurs temporarily but quickly. Such ‘‘pseudoergodicity’’ is also a symptom of chaos. We use the presence of quantum behavior that mimics local ergodicity and pseudoergodicity as a probe for the influence of classical chaos on quantum dynamics. We show that the quantum analog of a pseudoergodic region in phase space is generally formed by superpositions of energy eigenstates. The superposition states spanning a given pseudoergodic zone have similar expectation values and small off-diagonal elements to other states with similar energy for a certain class of operators. We perform calculations which identify the ergodic regions for two versions of the quantum Henon–Heiles system. For the ‘‘more classical’’ of these systems we find good agreement between the proportion of quantum states in pseudoergodic zones and the proportion of classical phase space occupied by chaotic trajectories. We also find that the quantum pseudoergodic regions can be identified with classical vague tori of the precessing and librating types. Different ergodic regions are separated from one another by what appear to be the quantum analogs of the precessor–librator separatrix and other classical bottlenecks. For the ‘‘less classical’’ of the systems, we find that classical chaos is reflected in the quantum dynamics to a smaller, but still noticeable, degree.

Multiplication of fractional calculus operators and boundary value problems involving the Euler-Darboux equation

With a view to presenting solutions of various boundary value problems involving the celebrated Euler-Darboux equation, the authors consider the multiplication of certain classes of operators of fractional calculus defined in terms of the Gaussian hypergeometric function. The fractional calculus operators studied here incorporate, as their special cases, both the Riemann-Liouville and Erdélyi-Kober operators, and are appropriately restricted in order to yield explicit solutions of some of the aforementioned boundary value problems in terms of Appell functions and Kampé de Fériet functions of two variables.

A note onC P estimates for certain kernels

For a certain class of operators defined by integral kernels, a necessary and sufficient condition is given for the belonging to the Schatten-von Neumann idealsC P. The operators considered generalize the classical Hankel operators; the results thus extend Peller's characterization of the Hankel operators in a classC P.