Sequence Of Linear Operators Research Articles

One-Sided Hyperbolicity via Evolution Maps

We give characterizations of the hyperbolicity of a nonautonomous one-sided dynamics defined by a sequence of linear operators, in terms of the hyperbolicity of an associated evolution map defined on a Banach space of sequences. We consider a large family of spaces of sequences besides $$L^p$$ spaces. Moreover, we discuss the general cases when the dynamics may be invertible only along the unstable direction and of a general nonuniform exponential behavior of the dynamics given by a sequence of norms instead of a single norm.

Admissibility and nonuniform polynomial dichotomies

AbstractFor a general one‐sided nonautonomous dynamics defined by a sequence of linear operators, we consider the notion of a polynomial dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility of bounded solutions. As a nontrivial application, we establish the robustness of the notion of a nonuniform polynomial dichotomy.

Sequence of Linear Operators in Non-Archimedean Banach Spaces

The aim of this paper is to find a non-Archimedean counterpart of the generalized convergence of closable unbounded linear operators as defined by Kato (Perturbation Theory for Linear Operators, 2nd edn. In: Grundlehren der Mathematischen Wissenschaften, Band 132, Springer, Berlin, 1976). Moreover, we prove that this convergence can be considered as a generalization of convergence in norm for unbounded linear operators on non-Archimedean Banach spaces (see Theorem 3.8).

A General Korovkin Result Under Generalized Convergence

In this paper the classic result of Korovkin about the convergence of sequences of functions defined from sequences of linear operators is reformulated in terms of generalized convergence. This convergence extends some others given in the literature. The operator of the sequence fulfill a shape preserving property more general than the positivity. This property is related with certain extension of the notion of derivative. This extended derivative is precisely the object of the approximation process. The study is completed by analysing the conditions for the existence of an asymptotic formula, from which some interesting consequences are derived as a local version of the shape preserving property. Finally, as applications of the previous results, the author use the following notion of generalized convergence, an extension of Nörlund-Cesaro summability given by V. Loku and N. L. Braha in 2017. A way to transfer a notion of generalized convergence to approximation theory by means of linear operators is showed.

High order approximation theory for Banach space valued functions

Here we study quantitatively the high degree of approximation of sequences of linear operators acting on Banach space valued di§erentiable functions to the unit operator. These operators are bounded by real positive linear companion operators. The Banach spaces considered here are general and no positivity assumption is made on the initial linear operators whose we study their approximation properties. We derive pointwise and uniform estimates which imply the approximation of these operators to the unit assuming di§erentiability of functions. At the end we study the special case where the high order derivative of the on hand function fulÖlls a convexity condition resulting into sharper estimates.
 MR3724631

SOLUTION OF NON-LINEAR PROBLEMS OF STRUCTURAL MECHANICS BY METHOD OF STEEPEST DESCENT

The algorithm of application of the method of steepest descent to the solution of problems of structural mechanics and solid mechanics, described by nonlinear differential equations. For application of this method to nonlinear operators are described by a sequence of linear operators in incremental form, unlimited and complex linear operator, in line with the idea of L. V. Kantorovich is limited to a simple linear unbounded operator.

Multivariate and abstract approximation theory for Banach space valued functions

Abstract Here we study quantitatively the high degree of approximation of sequences of linear operators acting on Banach space valued Fréchet differentiable functions to the unit operator, as well as other basic approximations including those under convexity. These operators are bounded by real positive linear companion operators. The Banach spaces considered here are general and no positivity assumption is made on the initial linear operators for which we study their approximation properties. We derive pointwise and uniform estimates, which imply the approximation of these operators to the unit assuming Fréchet differentiability of functions, and then we continue with basic approximations. At the end we study the special case where the approximated function fulfills a convexity condition resulting into sharp estimates. We give applications to Bernstein operators.

Feller’s Scheme in Approximation by Nonlinear Possibilistic Integral Operators

ABSTRACTBy analogy with Feller’s general probabilistic scheme used in the construction of many classical convergent sequences of linear operators, in this paper, we consider a Feller-kind scheme based on the possibilistic integral, for the construction of convergent sequences of nonlinear operators. In particular, in the discrete case, all the so-called max-product Bernstein-type operators and their qualitative convergence properties are recovered. Also, discrete nonperiodic nonlinear possibilistic convergent operators of Picard type, Gauss–Weierstrass type and Poisson–Cauchy type are studied and the possibility of introduction of discrete periodic(trigonometric) nonlinear possibilistic operators of de la Vallée–Poussin type, of Fejér type and of Jackson type is mentioned as future directions of research.

Hyperbolic Sequences of Linear Operators and Evolution Maps

For a nonautonomous dynamics defined by a sequence of linear operators on a Banach space, we give a characterization of its nonuniform hyperbolicity in terms of the hyperbolicity of an associated evolution map. We also obtain a generalization of this characterization to the class of nonuniformly hyperbolic cocycles over a locally compact topological space with values on the set of bounded linear operators.

Characterization of Nonuniform Contractions and Expansions with Growth Rates

For a one-sided nonautonomous dynamics defined by a sequence of linear operators, we obtain a complete characterization of (strong) nonuniform exponential contractions and (strong) nonuniform exponential expansions in terms of admissibility of certain pairs of sequence spaces. We allow asymptotic rates of the form \({e^{c\rho(n)}}\) determined by an arbitrary increasing sequence \({\rho(n)}\) that tends to infinity. For example, the usual exponential behavior with \({\rho(n) = n}\) is included as a very special case. As a nontrivial application of our work, we establish the robustness of (strong) nonuniform exponential contractions and (strong) nonuniform exponential expansions, that is, the persistence of those notions under sufficiently small linear perturbations.

Lyapunov theorems for exponential dichotomies in Hilbert spaces

For a nonautonomous dynamics defined by a sequence of linear operators, we obtain a complete characterization of the notion of a uniform exponential dichotomy in terms of the existence of appropriate Lyapunov sequences. In sharp contrast to previous results, we consider the case of noninvertible dynamics, thus requiring only the invertibility of operators along the unstable direction. Furthermore, we deal with operators acting on an arbitrary Hilbert space. As a nontrivial application of our work, we study the persistence of uniform exponential behavior under small linear and nonlinear perturbations.

Nonuniform hyperbolicity and one-sided admissibility

For a general one-sided nonautonomous dynamics defined by a sequence of linear operators, we consider the notion of an exponential dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility of bounded solutions. As a nontrivial application, we establish the robustness of the notion under sufficiently small parameterized perturbations. Moreover, we show that if the perturbations are Lipschitz or of class C^1 on the parameter, then the same happens to the projections onto the stable spaces of the perturbation.

Nonuniform exponential stability and admissibility

For cocycles defined by sequences of linear operators in a Banach space, we study the relation between the notions of nonuniform exponential stability and admissibility. The latter refers to the existence of bounded solutions for any bounded nonlinear perturbation of the original cocycle. In particular, using appropriate adapted norms to deal with the nonuniform behaviour, we show that if is admissible for a given cocycle, then the cocycle is a nonuniform exponential contraction. We also exhibit a collection of admissible Banach spaces for any given nonuniform exponential contraction. In addition, we obtain related results in the more general case of nonuniform exponential dichotomies. As an application of the ideas and methods discussed in the paper, we establish the robustness of nonuniform exponential contractions, in the sense that a sufficiently small linear perturbation of a nonuniform exponential contraction is again a nonuniform exponential contraction.

Admissibility, a general type of Lipschitz shadowing and structural stability

For a general one-sided nonautonomous dynamics defined by a sequence of linear operators, we consider the notion of a uniform exponential dichotomy and we characterize it completely in terms of the admissibility of a large class of function spaces. We apply those results to show that structural stability of a diffeomorphism is equivalent to a very general type of Lipschitz shadowing property. Our results extend those in [37] in various directions.

Characterization of strong exponential dichotomies

We introduce the notion of an exponential dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility of bounded solutions. The latter refers to the existence of (unique) bounded solutions for any bounded perturbation of the original dynamics. We consider the general case of a nonautonomous dynamics defined by a sequence of linear operators. As a nontrivial application, we establish the robustness of nonuniform exponential dichotomies as well as of strong nonuniform exponential dichotomies, which corresponds to the persistence of these notions under sufficiently small linear perturbations. The relevance of the results stems from the ubiquity of this type of exponential behavior in the context of ergodic theory: for almost all trajectories with nonzero Lyapunov exponents of a measure-preserving diffeomorphism, the derivative cocycle admits a nonuniform exponential dichotomy and in fact a strong nonuniform exponential dichotomy.

Nonuniform Hyperbolicity and Admissibility

Abstract For a nonautonomous dynamics defined by a sequence of linear operators, we consider the notion of an exponential dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility in pairs of spaces (ℓp, ℓq), with p and q not necessarily equal. This includes the notion of a nonuniform exponential dichotomy as a very special case. Moreover, we consider a general noninvertible dynamics and a strong nonuniform exponential behavior. The latter is the typical situation from the point of view of smooth ergodic theory.

Exponential dichotomies with respect to a sequence of norms and admissibility

For a nonautonomous dynamics defined by a sequence of linear operators, we introduce the notion of an exponential dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility in lp spaces, both for the space of perturbations and the space of solutions. This allows unifying the notions of uniform and nonuniform exponential behavior. Moreover, we consider the general case of a noninvertible dynamics. As a nontrivial application we show that the conditional stability of a nonuniform exponential dichotomy persists under sufficiently small linear perturbations.

Approximation of analytic functions by sequences of linear operators

In the present paper, we investigate approximation of analytic functions and their derivatives in a bounded simply connected domain by the sequences of linear operators without the properties of k?positivity.

On the β-Dual of Banach-Space-Valued Difference Sequence Spaces

The main object of the paper is to introduce Banach-space-valued difference sequence spaces l ∞(X, Δ), c(X, Δ); and c0(X, Δ) as a generalization of the well-known difference sequence spaces of Kizmaz. We obtain a set of sufficient conditions for (A k ) ∈ E β (X, Δ); where E ∈ {l ∞, c, c 0} and (A k ) is a sequence of linear operators from a Banach space X into another Banach space Y: Necessary conditions for (A k ) ∈ E β (X, Δ) are also investigated.

On a Double Complex Sequence of Linear Operators

The main goal of the article is to introduce a class of double complex linear operators of integral type. The technique is based by extension into the complex domain of a real positive approximation process. Involving the first modulus of continuity, we investigate their geometric and approximation properties. The statistical convergence of our sequence is proved. In a particular case, our operators turn into the double complex Gauss-Weierstrass integral operators.