scholarly journals Uniform Convergence and Spectra of Operators in a Class of Fréchet Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Angela A. Albanese ◽  
José Bonet ◽  
Werner J. Ricker
Keyword(s):  
Banach Space ◽  
Uniform Convergence ◽  
Bounded Operator ◽  
Operator Norm ◽  
Ergodic Theorems ◽  
Fréchet Spaces ◽  
Norm Convergence ◽  
Operator Ideals ◽  
Weakly Compact ◽  
Continuous Operators

Well-known Banach space results (e.g., due to J. Koliha and Y. Katznelson/L. Tzafriri), which relate conditions on the spectrum of a bounded operatorTto the operator norm convergence of certain sequences of operators generated byT, are extended to the class of quojection Fréchet spaces. These results are then applied to establish various mean ergodic theorems for continuous operators acting in such Fréchet spaces and which belong to certain operator ideals, for example, compact, weakly compact, and Montel.

scholarly journals The Dunford-Pettis property on vector-valued continuous and bounded functions

1993 ◽  
Vol 48 (2) ◽  
pp. 303-311 ◽  
Author(s):  
Jose Aguayo ◽  
Jose Sanchez
Keyword(s):  
Banach Space ◽  
Regular Space ◽  
Weakly Compact ◽  
Completely Regular ◽  
Bounded Functions ◽  
Continuous Operators ◽  
Vector Valued

Let X be a completely regular space, E a Banach space, Cb(X, E) the space of all continuous, bounded and E-valued functions defined on X, M(X, L(E, F)) the space of all L(E, F)-valued measures defined on the algebra generated by zero subsets of X. Weakly compact and β0-continuous operators defined from Cb(X, E) into a Banach space F are represented by integrals with respect to L(E, F)-valued measures. The strict Dunford-Pettis and the Dunford-Pettis properties are established on (Cb(X, E), βi), where βi denotes one of the strict topologies β0, β or β1, when E is a Schur space; the same properties are established on (Cb(X, E), β0), when E is an AM-space or an AL-space.

scholarly journals Polynomial Grothendieck properties

1995 ◽  
Vol 37 (2) ◽  
pp. 211-219 ◽  
Author(s):  
Manuel González ◽  
Joaquí M. Gutiérrez
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Homogeneous Polynomial ◽  
Bounded Operator ◽  
Weakly Compact ◽  
Completely Continuous ◽  
Projective Tensor Product ◽  
Linear Bounded Operator ◽  
Grothendieck Property ◽  
Projective Tensor

AbstractA Banach space sE has the Grothendieck property if every (linear bounded) operator from E into c0 is weakly compact. It is proved that, for an integer k > 1, every k-homogeneous polynomial from E into c0 is weakly compact if and only if the space (kE) of scalar valued polynomials on E is reflexive. This is equivalent to the symmetric A>fold projective tensor product of £(i.e., the predual of (kE)) having the Grothendieck property. The Grothendieck property of the projective tensor product EF is also characterized. Moreover, the Grothendieck property of E is described in terms of sequences of polynomials. Finally, it is shown that if every operator from E into c0 is completely continuous, then so is every polynomial between these spaces.

scholarly journals An operator satisfying Dunford's condition (C) but without bishop's property (β)

1998 ◽  
Vol 40 (3) ◽  
pp. 427-430 ◽  
Author(s):  
T. L. Miller ◽  
V. G. Miller
Keyword(s):  
Banach Space ◽  
Uniform Convergence ◽  
Complex Plane ◽  
Open Subset ◽  
Bounded Operator ◽  
Continuous Mapping ◽  
Complex Banach Space ◽  
Condition C ◽  
Analytic Subspace ◽  
Compact Subsets

For X a complex Banach space and U an open subset of the complex plane С, let O (U, X) denote the space of analytic X- valued functions defined on U. This is a Frechet space when endowed with the topology of uniform convergence on compact subsets, and the space X may be viewed as simply the constants in O(U, X). Every bounded operator T on X induces a continuous mapping TU on O(U, X) given by (Tuf)(λ) = (λ – T)f(λ) for every f e O(U, X) and λ e U. Corresponding to each closed F ⊂ С there is also an associated analytic subspace XT(F) = X ∩ ran(7c//F). For an arbitrary T e L(X), the spaces XT(F) are T-invariant, generally non-closed linear manifolds in X.

scholarly journals Convergence of generic infinite products of affine operators

1999 ◽  
Vol 4 (1) ◽  
pp. 1-19 ◽  
Author(s):  
S. Reich ◽  
A. J. Zaslavski
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Asymptotic Behavior ◽  
Common Fixed Point ◽  
Ergodic Theorems ◽  
Infinite Products ◽  
Continuous Operators ◽  
Uniformly Continuous ◽  
Unique Common Fixed Point ◽  
Convex Subsets

We establish several results concerning the asymptotic behavior of random infinite products of generic sequences of affine uniformly continuous operators on bounded closed convex subsets of a Banach space. In addition to weak ergodic theorems we also obtain convergence to a unique common fixed point and more generally, to an affine retraction.

scholarly journals The compact weak topology on a Banach space

1992 ◽  
Vol 120 (3-4) ◽  
pp. 367-379 ◽  
Author(s):  
Manuel González ◽  
Joaquín M. Gutiérrez
Keyword(s):  
Banach Space ◽  
Uniform Convergence ◽  
Banach Spaces ◽  
Weak Topology ◽  
Dual Space ◽  
Holomorphic Maps ◽  
Weakly Compact ◽  
Natural Manner ◽  
Locally Convex Topology ◽  
Locally Convex

SynopsisThe compact weak topology (kw) on a Banach space is defined as the finest topology that agrees with the weak topology on weakly compact subsets. It appears in a natural manner in the study of certain classes of continuous and holomorphic maps between Banach spaces. In this paper we treat the kw topology and the finest locally convex topology contained in kw, which we call the ckw topology. We prove that kw = ckw if and only if the space is reflexive or Schur, and we derive characterisations of Banach spaces not containing l1, and of other classes of Banach spaces, in terms of these topologies. We also show that ckw is the topology of uniform convergence on (L)-subsets of the dual space. As a consequence, Banach spaces with the reciprocal Dunford–Pettis property are characterised.

scholarly journals Banach spaces with property (w)

1993 ◽  
Vol 35 (2) ◽  
pp. 207-217 ◽  
Author(s):  
Denny H. Leung
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Banach Lattice ◽  
Banach Lattices ◽  
Weakly Compact ◽  
Type Spaces

A Banach space E is said to have Property (w) if every operator from E into E' is weakly compact. This property was introduced by E. and P. Saab in [9]. They observe that for Banach lattices, Property (w) is equivalent to Property (V*), which in turn is equivalent to the Banach lattice having a weakly sequentially complete dual. Thus the following question was raised in [9].Does every Banach space with Property (w) have a weakly sequentially complete dual, or even Property (V*)?In this paper, we give two examples, both of which answer the question in the negative. Both examples are James type spaces considered in [1]. They both possess properties stronger than Property (w). The first example has the property that every operator from the space into the dual is compact. In the second example, both the space and its dual have Property (w). In the last section we establish some partial results concerning the problem (also raised in [9]) of whether (w) passes from a Banach space E to C(K, E).

Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions

1999 ◽  
Vol 42 (2) ◽  
pp. 139-148 ◽  
Author(s):  
José Bonet ◽  
Paweł Dománski ◽  
Mikael Lindström
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Composition Operators ◽  
Essential Norm ◽  
Weak Compactness ◽  
Weakly Compact ◽  
Point Evaluation ◽  
Weighted Banach Spaces ◽  
Spaces Of Analytic Functions

AbstractEvery weakly compact composition operator between weighted Banach spaces of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space are also estimated, thus permitting to get new characterizations of compact composition operators between these spaces.

CLASSES OF SEQUENTIALLY LIMITED OPERATORS

2015 ◽  
Vol 58 (3) ◽  
pp. 573-586
Author(s):  
JAN H. FOURIE ◽  
ELROY D. ZEEKOEI
Keyword(s):  
Banach Space ◽  
Vector Space ◽  
Normed Space ◽  
Operator Ideal ◽  
Operator Ideals ◽  
Ideal Norm ◽  
Relevant Operator

AbstractThe purpose of this paper is to present a brief discussion of both the normed space of operator p-summable sequences in a Banach space and the normed space of sequentially p-limited operators. The focus is on proving that the vector space of all operator p-summable sequences in a Banach space is a Banach space itself and that the class of sequentially p-limited operators is a Banach operator ideal with respect to a suitable ideal norm- and to discuss some other properties and multiplication results of related classes of operators. These results are shown to fit into a general discussion of operator [Y,p]-summable sequences and relevant operator ideals.

On the contractibility to a point of the linear groups of reflexive non-commutative Lp-spaces

1996 ◽  
Vol 119 (3) ◽  
pp. 545-560 ◽  
Author(s):  
Sergei V. Ferleger ◽  
Fyodor A. Sukochev
Keyword(s):  
Banach Space ◽  
Linear Group ◽  
Operator Norm ◽  
Linear Groups ◽  
Lp Spaces ◽  
Continuous Map

For every Banach space X, denote by GL(X) the linear group of X, i.e. the group of all linear continuous invertible operators on X with the topology induced by the operator norm. One says that GL(X) is contractible to a point if there exists a continuous map F: GL(X) × [0, 1] → GL(X) such that F(A,0) = A and F(A, 1) = Id, for every A ∈ GL(X).

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