Power Boundedness Research Articles

Spectra and dynamics of generalized Cesàro operators in (LF) and (PLB) sequence spaces

In this paper, we introduce inductive limits of the Fréchet spaces ℓ(p+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell (p+)$$\\end{document}, ces(p+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {ces}(p+)$$\\end{document}, and d(p+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d(p+)$$\\end{document} (1≤p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 \\le p < \\infty $$\\end{document}) and projective limits of the (LB)-spaces ℓ(p-)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell (p-)$$\\end{document}, ces(p-)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {ces}(p-)$$\\end{document}, and d(p-)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d(p-)$$\\end{document} (1<p≤∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 < p \\le \\infty $$\\end{document}). After having established some topological properties of such spaces as acyclicity and ultrabornologicity, we prove that the generalized Cesàro operators Ct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_t$$\\end{document} (0≤t≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0 \\le t \\le 1$$\\end{document}) act continuously in these sequence spaces, and we determine the spectra. Finally, we study the ergodic properties, that is, power boundedness, (uniform) mean ergodicity, and supercyclicity, of the operators Ct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_t$$\\end{document} acting in the (LF)-spaces and in the (PLB)-spaces mentioned above.

Ergodicity for a family of operators

The aim of this paper is to introduce the notions of power boundedness, Cesàro boundedness, mean ergodicity, and uniform ergodicity for a family of bounded linear operators on a Banach space. The authors present some elementary results in this setting and show that some main results about power bounded, Cesàro bounded, mean ergodic, and the uniform ergodic operator can be extended from the case of a linear bounded operator to the case of a family of bounded linear operators acting on a Banach space. Also, we show that the Yosida theorem can be extended from the case of a bounded linear operator to the case of a family of bounded linear operators acting on a Banach space.

Mean Ergodic Weighted Shifts on Köthe Echelon Spaces

Necessary and sufficient conditions are given for mean ergodicity, power boundedness, and topologizability for weighted backward shift and weighted forward shift operators, respectively, on Köthe echelon spaces in terms of the weight sequence and the Köthe matrix. These conditions are evaluated for the special case of power series spaces which allow for a characterization of said properties in many cases. In order to demonstrate the applicability of our conditions, we study the above properties for several classical operators on certain function spaces.

Plane-Wave Diffraction from Resistive-Filled Circular Hole in Infinite Resistive Plane: An Analytically Regularizing Approach

The study of the electromagnetic diffraction from penetrable screens with apertures and/or inhomogeneities is of great relevance today due to the huge number of modern applications in which they are involved. In this paper, the analysis of the plane wave scattering from a resistive-filled circular hole in a resistive plane is addressed. The uniquely solvable boundary value problem for the Maxwell equations, obtained via imposing generalized boundary conditions, power boundedness condition, and Silver–Muller radiation condition, is equivalently formulated in terms of an infinite set of singular dual integral equations in the vector Hankel transform domain. The Helmholtz–Galerkin technique allows for the discretization and, simultaneously, analytical regularization of the obtained integral equations. Fast convergence is guaranteed by a suitable choice of the basis functions reconstructing the physical behavior of the fields at the discontinuity between the two involved media. Moreover, the full-wave nature of the proposed approach allows the direct assessment of near-field and far-field parameters.

Liouville closed HT-fields

Let T be a complete, model complete o-minimal theory extending the theory of real closed ordered fields. An HT-field is a model K of T equipped with a T-derivation ▪ such that the underlying ordered differential field of ▪ is an H-field. We study HT-fields and their extensions. Our main result is that if T is power bounded, then every HT-field K has either exactly one or exactly two minimal Liouville closed HT-field extensions up to K-isomorphism. The assumption of power boundedness can be relaxed to allow for certain exponential cases, such as T=Th(Ran,exp).

T-convex T-differential fields and their immediate extensions

Let $T$ be a polynomially bounded o-minimal theory extending the theory of real closed ordered fields. Let $K$ be a model of $T$ equipped with a $T$-convex valuation ring and a $T$-derivation. If this derivation is continuous with respect to the valuation topology, then we call $K$ a $T$-convex $T$-differential field. We show that every $T$-convex $T$-differential field has an immediate strict $T$-convex $T$-differential field extension which is spherically complete. In some important cases, the assumption of polynomial boundedness can be relaxed to power boundedness.

Frequently recurrent operators

Motivated by a recent investigation of Costakis et al. on the notion of recurrence in linear dynamics, we study various stronger forms of recurrence for linear operators, in particular that of frequent recurrence. We study, among other things, the relationship between a type of recurrence and the corresponding notion of hypercyclicity, the influence of power boundedness, and the interplay between recurrence and spectral properties. We obtain, in particular, Ansari- and Léon-Müller-type theorems for F-recurrence under very weak assumptions on the Furstenberg family F. This allows us, as a by-product, to deduce Ansari- and Léon-Müller-type theorems for F-hypercyclicity.

On diagonal operators between the sequence (LF)-spaces l_p({mathcal {V}})

Diagonal (multiplication) operators acting between a particular class of countable inductive spectra of Fréchet sequence spaces, called sequence (LF)-spaces, are investigated. We prove results concerning boundedness, compactness, power boundedness, and mean ergodicity. Furthermore, we determine when a diagonal operator is Montel and reflexive. We analyze the spectra in terms of the system of weights defining the spaces.

Dynamics of composition operators on function spaces defined by local and global properties

In this paper we consider composition operators on locally convex spaces of functions defined on R. We prove results concerning supercyclicity, power boundedness, mean ergodicity and convergence of the iterates in the strong operator topology.

Sharp stability for finite difference approximations of hyperbolic equations with boundary conditions

Abstract In this article, we consider a class of finite rank perturbations of Toeplitz operators that have simple eigenvalues on the unit circle. Under a suitable assumption on the behavior of the essential spectrum, we show that such operators are power bounded. The problem originates in the approximation of hyperbolic partial differential equations with boundary conditions by means of finite difference schemes. Our result gives a positive answer to a conjecture by Trefethen, Kreiss and Wu that only a weak form of the so-called uniform Kreiss–Lopatinskii condition is sufficient to imply power boundedness.

Dynamics of Weighted Composition Operators on Spaces of Entire Functions of Exponential and Infraexponential Type

Given an affine symbol varphi and a multiplier w, we focus on the weighted composition operator C_{w, varphi } acting on the spaces Exp and Exp^0 of entire functions of exponential and of infraexponential type, respectively. We characterize the continuity of the operator and, for w the product of a polynomial by an exponential function, we completely characterize power boundedness and (uniform) mean ergodicity. In the case of multiples of composition operators, we also obtain the spectrum and characterize hypercyclicity.

Power boundedness in the maximum norm of stability matrices for ADI methods

Power boundedness in the maximum norm of stability matrices for ADI methods

On the Ritt condition on Locally Convex Vector Spaces

In this paper, we show that the Ritt condition in the case of locally convex spaces can be related to the power boundedness of a universally bounded operator. We will characterize this condition by two geometric properties of the powers and we prove that the Ritt condition will be shown to be equivalent to the Tadmor condition. We study the Ritt condition for a quasinilpotent operator acting on locally convex spaces. Also, an upper bound for the norm of the powers of operators acting on locally convex spaces under Ritt condition was given. <br><br> Показано, що у випадку локально опуклих просторів умова Рітта пов’язана з обмеженістю степенів універсально обмеженого оператора. Ця умова характеризується в термінах геометричних властивостей степенів. Доведено, що умова Рітта еквівалентна умові Тедмора. Досліджена умова Рітта для вирадку квазінільпотентних операторів у локально опуклих просторах. Знайдена також верхня оцінка норм степенів операторів, які задовольняють умову Рітта.

Dynamics of weighted composition operators on weighted Banach spaces of entire functions

We study the dynamics of the weighted composition operator Cw,φ on weighted Banach spaces of entire functions Hv(C) and Hv0(C). We characterize the continuity and compactness of the operator and, in the case of affine symbols φ(z)=az+b,a,b∈C, and exponential weights, we analyze when the operator is power bounded, (uniformly) mean ergodic and hypercyclic. Continuous weighted composition operators when |a|=1 are just multiples of composition operators λCφλ∈C. When |a|<1, we consider as a multiplier w the product of a polynomial by an exponential function. For multiples of composition operators, we get a complete characterization of power boundedness and mean ergodicity and we study the hypercyclicity in terms of λ. An example of a power bounded but not mean ergodic operator on Hv0(C) is provided. For the case of composition operators, we obtain the spectrum and a complete characterization of the dynamics.

Topologizable and power bounded weighted composition operators on spaces of distributions

We study topologizability and power boundedness of weighted composition operators on (certain subspaces of) $\mathscr {D}’(X)$ for an open subset $X$ of $\mathbb {R}^d$. For the former property we derive a characterization in terms of the symbol and the w

Power bounded weighted composition operators on function spaces defined by local properties

We study power boundedness and related properties such as mean ergodicity for (weighted) composition operators on function spaces defined by local properties. As a main application of our general approach we characterize when (weighted) composition operators are power bounded, topologizable, and (uniformly) mean ergodic on kernels of certain linear partial differential operators including elliptic operators as well as non-degenrate parabolic operators. Moreover, under mild assumptions on the weight and the symbol we give a characterisation of those weighted composition operators on the Fr\'echet space of continuous functions on a locally compact, $\sigma$-compact, non-compact Hausdorff space which are generators of strongly continuous semigroups on these spaces.

A Sufficient Condition for the Similarity of a Polynomially Bounded Operator to a Contraction

Let $T$ be a polynomially bounded operator, and let $\mathcal M$ be its invariant subspace. Suppose that $P_{\mathcal M^\perp}T|_{\mathcal M^\perp}$ is similar to a contraction, while $\theta(T|_{\mathcal M})=0$, where $\theta$ is a finite product of Blaschke products with simple zeros satisfying the Carleson interpolating condition (a Carleson--Newman Blaschke product). Then $T$ is similar to a contraction. It is mentioned that Le Merdy's example shows that the assumption of polynomially boundedness cannot be replaced by the assumption of power boundedness.

Microspectral analysis of quasinilpotent operators

We develop a microspectral theory for quasinilpotent linear operators Q (i.e., those with σ(Q) = {0}) in a Banach space. When such Q is not compact, normal, or nilpotent, the classical spectral theory gives little information, and a somewhat deeper structure can be recovered from microspectral sets in C. Such sets describe, e.g., semigroup generation, resolvent properties, power boundedness as well as Tauberian properties associated to zQ for z ∈ C.

Power boundedness in the Fourier and Fourier–Stieltjes algebras on homogeneous spaces

We study power boundedness in the Fourier and Fourier–Stieltjes algebras, Open image in new window and Open image in new window of a homogeneous space Open image in new window The main results characterizes when all elements with spectral radius at most one, in any of these algebras, are power bounded.

Mean ergodicity of weighted composition operators on spaces of holomorphic functions

Let φ be a self-map of the unit disc D of the complex plane C and let ψ be a holomorphic function on D. We investigate the mean ergodicity and power boundedness of the weighted composition operator Cφ,ψ(f)=ψ(f∘φ) with symbol φ and multiplier ψ on the space H(D). We obtain necessary and sufficient conditions on the symbol φ and on the multiplier ψ which characterize when the weighted composition operator is power bounded and (uniformly) mean ergodic. One necessary condition is that the symbol φ has a fixed point in D. If φ is not a rational rotation, the sufficient conditions are related to the modulus of the multiplier on the fixed point of φ. Some of our results are valid in an open connected set U of the complex plane.