Localizable spectrum and bounded local resolvent functions

Given a Banach space operator with interior points in the localizable spectrum and without non-trivial divisible subspaces, this article centers around the construction of an infinite-dimensional linear subspace of vectors at which the local resolvent function of the operator is bounded and even admits a continuous extension to the closure of its natural domain. As a consequence, it is shown that, for any measure with natural spectrum on a locally compact abelian group, the corresponding operator of convolution on the group algebra admits a non-zero bounded local resolvent function precisely when its spectrum has non-empty interior. Received: 15 November 2007     

Subexponential decay of correlations

We describe a method for proving subexponential lower bounds for correlations functions, and apply it to study decay of correlations for maps with countable Markov partitions. One result is that LS Young’s upper estimates [Y2] are optimal in many situations. Our method is based on a general result concerning the asymptotics of renewal sequences of bounded operators acting on Banach spaces, which we apply to the iterates of the transfer operator. Oblatum 14-VII-2001 & 5-VI-2002?Published online: 6 August 2002 Dedicated to B. Weiss on the occasion of his 60th birthday RID="*" ID="*"This research was supported by the Nuffield Foundation, NUF-NAL 00.     

Banach Lattices on Which Every Power-Bounded Operator is Mean Ergodic

Given a Banach lattice E that fails to be countably order complete, we construct a positive compact operator A: E E for which T = I - A is power-bounded and not mean ergodic. As a consequence, by using the theorem of R. Zaharopol, we obtain that if every power-bounded operator in a Banach lattice is mean ergodic then the Banach lattice is reflexive.     

Weak Module Amenability for Semigroup Algebras

We study the weak module amenability of Banach algebras which are Banach module over another Banach algebra with compatible actions. As an example we show that the semigroup algebra of a commutative inverse semigroup is always weakly amenable as a module over the semigroup algebra of its subsemigroup of idempotents.     

Isolated points and essential components of composition operators on

We consider the topological space of all composition operators on the Banach algebra of bounded analytic functions on the unit disk. We obtain a function theoretic characterization of isolated points and show that each isolated composition operator is essentially isolated.

     

A generalized notion of variation applied to Markov chains and Anosov maps

Summary Extending the operator formalism of [3] we show that there exists a large class of functions which possess an exponential decay of correlations and fulfill a central limit theorem under a certain type of Markov chains. This result can be applied to the symbolic dynamics of Anosov maps, showing that in the case of a absolutely continuous invariant measure there is a large class of functions with good ergodic properties-larger than the usual class of Hölder continuous functions.work supported by Studienstiftung des deutschen Volkes     

A quantitative nonlinear strong ergodic theorem for Hilbert spaces

We give a quantitative version of a strong nonlinear ergodic theorem for (a class of possibly even discontinuous) selfmappings of an arbitrary subset of a Hilbert space due to R. Wittmann and outline how the existence of uniform bounds in such quantitative formulations of ergodic theorems can be proved by means of a general logical metatheorem. In particular these bounds depend neither on the operator nor on the initial point. Furthermore, we extract such uniform bounds in our quantitative formulation of Wittmann?s theorem, implicitly using the proof-theoretic techniques on which the metatheorem is based. However, we present our result and its proof in analytic terms without any reference to logic as such. Our bounds turn out to involve nested iterations of relatively low computational complexity. While in theory these kind of iterations ought to be expected, so far this seems to be the first occurrence of such a nested use observed in practice.     

Module operator virtual diagonals on the Fourier algebra of an inverse semigroup

For an amenable inverse semigroup S with the set of idempotents E and a minimal idempotent, we explicitly construct a contractive and positive module operator virtual diagonal on the Fourier algebra A(S), as a completely contractive Banach algebra and operator module over \(\ell ^1(E)\). This generalizes a well known result of Zhong-Jin Ruan on operator amenability of the Fourier algebra of a (discrete) group Ruan (Am J Math 117:1449–1474, 1995).     

Algebra homomorphisms defined via convoluted semigroups and cosine functions

Transform methods are used to establish algebra homomorphisms related to convoluted semigroups and convoluted cosine functions. Such families are now basic in the study of the abstract Cauchy problem. The framework they provide is flexible enough to encompass most of the concepts used up to now to treat Cauchy problems of the first- and second-order in general Banach spaces. Starting with the study of the classical Laplace convolution and a cosine convolution, along with associated dual transforms, natural algebra homomorphisms are introduced which capture the convoluted semigroup and cosine function properties. These correspond to extensions of the Cauchy functional equation for semigroups and the abstract d'Alembert equation for the case of cosine operator functions. The algebra homomorphisms obtained provide a way to prove Hille-Yosida type generation theorems for the operator families under consideration.     

Banach空间的Lipschitz对偶及其应用

本文引进Banach空间E的一个全新对偶空间概念—Lipschitz对偶空间,并证明：任何Banach空间的Lipschitz对偶空间是某个包含E的Banach空间的线性对偶空间,以所引进的新对偶空间为框架,本文定义了非线性Lipschitz算子的Lipshitz对偶算子,证明：任何非线性Lipschitz算子的Lipschitz对偶算子是有界线性算子．所获结果为推广线性算子理论到非线性情形（特别,运用线性算子理论研究非线性算子的特性）开辟了一条新的途径．作为例证,我们应用所建立的理论证明了若干新的非线性一致Lipschitz映象遍历收敛性定理.     

Invariant measures for frequently hypercyclic operators

We investigate frequently hypercyclic and chaotic linear operators from a measure-theoretic point of view. Among other things, we show that any frequently hypercyclic operator T acting on a reflexive Banach space admits an invariant probability measure with full support, which may be required to vanish on the set of all periodic vectors for T  ; that there exist frequently hypercyclic operators on the sequence space _c0$c_0$ admitting no ergodic measure with full support; and that if an operator admits an ergodic measure with full support, then it has a comeager set of distributionally irregular vectors. We also give some necessary and sufficient conditions (which are satisfied by all the known chaotic operators) for an operator T to admit an invariant measure supported on the set of its hypercyclic vectors and belonging to the closed convex hull of its periodic measures. Finally, we give a Baire category proof of the fact that any operator with a perfectly spanning set of unimodular eigenvectors admits an ergodic measure with full support.     

On an algebra of multidimensional integral operators with homogeneous-difference kernels

A Banach algebra generated by multidimensional integral operators with homogeneous-difference kernels is considered. A symbolic calculus is constructed for this algebra and, in terms of this calculus, necessary and sufficient conditions for the invertibility of an operator are obtained.     

Finite sums of idempotents and logarithmic residues on connected domains

We show that every finite sum of idempotents in a Banach algebra can be represented as the logarithmic residue of some analytic Banach algebra valued function defined on any, given bounded Cauchy domain. Moreover, using this, we can construct a non-invertible analytic Banach algebra valued function which is defined on any given bounded Cauchy domain and whose logarithmic residue is equal to zero., Consequently, the classical theorem concerning logarithmic residues fails in the general situation for all domains, in particular for connected domains.     

Asymptotic behavior of solutions to nonlinear Volterra integral equations

A class of operator Riccati integral equations is associated with a factorization problem in a certain Banach algebra. Recent results concerning factorization in this algebra are used to obtain existence, uniqueness, and continuous dependence results for the Riccati equations.     

Intrinsic Localization of Frames

Several concepts for the localization of a frame are studied. The intrinsic localization of a frame is defined by the decay properties of its Gramian matrix. Our main result asserts that the canonical dual frame possesses the same intrinsic localization as the original frame. The proof relies heavily on Banach algebra techniques, in particular on recent spectral invariance properties for certain Banach algebras of infinite matrices. Intrinsically localized frames extend in a natural way to Banach frames for a class of associated Banach spaces which are defined by weighted ℓ^p-coefficients of their frame expansions. As an example, the time--frequency concentration of distributions is characterized by means of localized (nonuniform) Gabor frames.     

Nonlinear Ergodic Theorem for Positively Homogeneous Nonexpansive Mappings in Banach Spaces

Recently, two retractions (projections), which are different from the metric projection and the sunny nonexpansive retraction in a Banach space, were found. In this article, using nonlinear analytic methods and new retractions, we prove a nonlinear ergodic theorem for positively homogeneous and nonexpansive mappings in a uniformly convex Banach space. The limit points are characterized by using new retractions.     

Generation and commutation properties of the Volterra operator

Let V be the classical Volterra operator on L ₂(0,1). Then the algebra generated (algebraically) by V and its adjoint is not only dense in the Banach space of all compact operators, but also in the Banach space of all Hilbert?CSchmidt operators and as well in the space ${\mathcal{B}(L_2(0,1))}$ equipped with the weak operator topology. Moreover, the algebra generated by V ² and its adjoint is dense in the Banach space of all trace class operators. We give an elementary proof that similar results are valid for polynomials in V without constant term. We also show that the commutant of any non-constant analytic function of V coincides with the commutant of V.     

QUOTIENTS OF 2-SUMMING ALGEBRAS

Abstract

A result of A.M. Mantero and A. Tonge is strengthened by giving an example of a commutative operator algebra which is semisimple but which is not the quotient algebra of a 2-summing Banach algebra by a closed ideal.     

Gevrey Class Regularity and Exponential Decay Property for Navier-Stokes-α Equations

The Navier-Stokes-α equations subject to the periodic boundary conditions are considered.An-alyticity in time for a class of solutions taking values in a Gevrey class of functions is proven.Exponentialdecay of the spatial Fourier spectrum for the analytic solutions and the lower bounds on the rate defined by theexponential decay are also obtained.     

三角Banach代数的Lie弱顺从性

设$\mathcal {A,\ B}$ 是含单位元的Banach代数, $\mathcal M$ 是一个Banach $\mathcal {A,\ B}$-双模. $\mathcal {T}=\left ( \begin{array}{cc} \mathcal {A} & \mathcal M \\ & \mathcal {B} \\ \end{array} \right )$按照通常矩阵加法和乘法,范数定义为$\|\left( \begin{array}{cc} a & m \\ & b\\ \end{array} \right)\|=\|a\|_{\mathcal A}+\|m\|_{\mathcal M}+\|b\|_{\mathcal B}$,构成三角Banach 代数.如果从$\mathcal T$到其$n$次对偶空间$\mathcal T^{n}$上的Lie导子都是标准的,则称$\mathcal T$是Lie $n$弱顺从的.本文研究了三角Banach代数$\mathcal T$上的Lie $n$弱顺从性,证明了有限维套代数是Lie $n$弱顺从的.