scholarly journals On Boolean algebras of projections

1977 ◽  
Vol 18 (1) ◽  
pp. 69-72
Author(s):  
J. A. Erdos
Keyword(s):  
Banach Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Boolean Algebras ◽  
Alternative Methods ◽  
Weakly Compact ◽  
Bounded Linear ◽  
Turn On ◽  
Underlying Space ◽  
Unit Norm

In spectral theory on Banach spaces, certain more incisive results hold when the underlying space is weakly complete (that is, weakly sequentially complete). The standard proofs rely on the following deep theorem: any bounded linear map from the algebra of all complex continuous functions on a compact Hausdorff space to a weakly complete Banach space is weakly compact. The proof of this result depends in turn on a considerable amount of measure-theoretic machinery (see [4, Section VI.7]). We present here some alternative methods which avoid these technicalities. The results are then used to give an example of a set of projections, each having unit norm, which generate an unbounded Boolean algebra.

Characterization of some classes of operators on spaces of vector-valued continuous functions

1985 ◽  
Vol 97 (1) ◽  
pp. 137-146 ◽  
Author(s):  
Fernando Bombal ◽  
Pilar Cembranos
Keyword(s):  
Banach Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Supremum Norm ◽  
Representing Measure ◽  
Bounded Linear ◽  
Second Dual ◽  
Image Position ◽  
Vector Valued

Let K be a compact Hausdorff space and E, F Banach spaces. We denote by C(K, E) the Banach space of all continuous. E-valued functions defined on K, with the supremum norm. It is well known ([6], [7]) that every operator (= bounded linear operator) T from C(K, E) to F has a finitely additive representing measure m of bounded semi-variation, defined on the Borel σ-field Σ of K and with values in L(E, F″) (the space of all operators from E into the second dual of F), in such a way thatwhere the integral is considered in Dinculeanu's sense.

scholarly journals BOUNDED LINEAR OPERATORS ON SPACES IN NORMED DUALITY

2007 ◽  
Vol 49 (1) ◽  
pp. 145-154
Author(s):  
BRUCE A. BARNES
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Integral Operators ◽  
Continuous Functions ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Closed Range ◽  
Bounded Linear ◽  
Spaces Of Continuous Functions

Abstract.LetTbe a bounded linear operator on a Banach spaceW, assumeWandYare in normed duality, and assume thatThas adjointT†relative toY. In this paper, conditions are given that imply that for all λ≠0, λ−Tand λ −T†maintain important standard operator relationships. For example, under the conditions given, λ −Thas closed range if, and only if, λ −T†has closed range.These general results are shown to apply to certain classes of integral operators acting on spaces of continuous functions.

On Additive Operators

1971 ◽  
Vol 23 (3) ◽  
pp. 468-480 ◽  
Author(s):  
N. A. Friedman ◽  
A. E. Tong
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Weak Topology ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Additive Functional ◽  
Representation Theorems ◽  
Image Position ◽  
Vector Valued ◽  
Kernel Representation

Representation theorems for additive functional have been obtained in [2, 4; 6-8; 10-13]. Our aim in this paper is to study the representation of additive operators.Let S be a compact Hausdorff space and let C(S) be the space of real-valued continuous functions defined on S. Let X be an arbitrary Banach space and let T be an additive operator (see § 2) mapping C(S) into X. We will show (see Lemma 3.4) that additive operators may be represented in terms of a family of “measures” {μh} which take their values in X**. If X is weakly sequentially complete, then {μh} can be shown to take their values in X and are vector-valued measures (i.e., countably additive in the norm) (see Lemma 3.7). And, if X* is separable in the weak-* topology, T may be represented in terms of a kernel representation satisfying the Carathéordory conditions (see [9; 11; §4]):

scholarly journals The () Property in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Danyal Soybaş
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Isomorphic Copy ◽  
Weakly Compact ◽  
Geometric Properties ◽  
Bounded Linear ◽  
Real Functions

A Banach space is said to have (D) property if every bounded linear operator is weakly compact for every Banach space whose dual does not contain an isomorphic copy of . Studying this property in connection with other geometric properties, we show that every Banach space whose dual has (V∗) property of Pełczyński (and hence every Banach space with (V) property) has (D) property. We show that the space of real functions, which are integrable with respect to a measure with values in a Banach space , has (D) property. We give some other results concerning Banach spaces with (D) property.

scholarly journals STRONGLY BOUNDED REPRESENTING MEASURES AND CONVERGENCE THEOREMS

2010 ◽  
Vol 52 (3) ◽  
pp. 435-445 ◽  
Author(s):  
IOANA GHENCIU ◽  
PAUL LEWIS
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Supremum Norm ◽  
Convergence Theorems

AbstractLet K be a compact Hausdorff space, X a Banach space and C(K, X) the Banach space of all continuous functions f: K → X endowed with the supremum norm. In this paper we study weakly precompact operators defined on C(K, X).

scholarly journals Weakly compact operators and the strict topologies

1989 ◽  
Vol 39 (3) ◽  
pp. 353-359 ◽  
Author(s):  
José Aguayo ◽  
José Sánchez
Keyword(s):  
Banach Space ◽  
Compact Operators ◽  
Continuous Functions ◽  
Supremum Norm ◽  
Regular Space ◽  
Weakly Compact ◽  
Completely Regular ◽  
Weakly Compact Operators ◽  
Bounded Continuous Functions

Let X be a completely regular space. We denote by Cb(X) the Banach space of all real-valued bounded continuous functions on X endowed with the supremum-norm.In this paper we prove some characterisations of weakly compact operators defined from Cb(X) into a Banach space E which are continuous with respect to fit, βt, βr and βσ introduced by Sentilles.We also prove that (Cb,(X), βi), i = t, τσ , has the Dunford-Pettis property.

scholarly journals Factoring absolutely summing operators through Hilbert-Schmidt operators

1989 ◽  
Vol 31 (2) ◽  
pp. 131-135 ◽  
Author(s):  
Hans Jarchow
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hausdorff Space ◽  
Bounded Operator ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
Reflexive Banach Spaces ◽  
Summing Operator ◽  
Schmidt Operator ◽  
Second Dual

Let K be a compact Hausdorff space, and let C(K) be the corresponding Banach space of continuous functions on K. It is well-known that every 1-summing operator S:C(K)→l2 is also nuclear, and therefore factors S = S1S2, with S1:l2→l2 a Hilbert–Schmidt operator and S1:C(K)→l2 a bounded operator. It is easily seen that this latter property is preserved when C(K) is replaced by any quotient, and that a Banach space X enjoys this property if and only if its second dual, X**, does. This led A. Pełczyński [15] to ask if the second dual of a Banach space X must be isomorphic to a quotient of a C(K)-space if X has the property that every 1-summing operator X-→l2 factors through a Hilbert-Schmidt operator. In this paper, we shall first of all reformulate the question in an appropriate manner and then show that counter-examples are available among super-reflexive Tsirelson-like spaces as well as among quasi-reflexive Banach spaces.

scholarly journals A Remark on the Continuity of the Dual Process

1968 ◽  
Vol 32 ◽  
pp. 287-295 ◽  
Author(s):  
Mamoru Kanda
Keyword(s):  
Banach Space ◽  
Compact Support ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Finite Measure ◽  
Uniform Norm ◽  
Continuous Functions ◽  
Dual Process ◽  
Space Of Continuous Functions ◽  
Locally Compact

Let S be a locally compact (not compact) Hausdorff space satisfying the second axiom of countability and let ℬ be the σ field of all Borel subsets of S and let A be the σ-field of all the subsets of S which, for each finite measure μ defined on (S, A), are in the completed σ field of ℬ relative to μ. We denote by C0 the Banach space of continuous functions vanishing at infinity with the uniform norm and Bk the space of bounded A-measurable functions with compact support in S.

Weakly compact, unconditionally converging, and Dunford-Pettis operators on spaces of vector-valued continuous functions

1984 ◽  
Vol 95 (1) ◽  
pp. 101-108 ◽  
Author(s):  
Paulette Saab
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Representing Measure ◽  
Weakly Compact ◽  
Bounded Linear ◽  
Borel Set ◽  
Nikodym Property ◽  
Weakly Compact Operators

Given a compact Hausdorff space X, E and F two Banach spaces, let T: C(X, E) → F denote a bounded linear operator (here C(X, E) stands for the Banach space of all continuous E-valued functions defined on X under supremum norm). It is well known [4] that any such operator T has a finitely additive representing measure G that is defined on the σ–field of Borel subsets of X and that G takes its values in the space of all bounded linear operators from E into the second dual of F. The representing measure G enjoys a host of many important properties; we refer the reader to [4] and [5] for more on these properties. The question of whether properties of the operator T can be characterized in terms of properties of the representing measure has been considered by many authors, see for instance [1], [2], [3] and [6]. Most characterizations presented (see [3] concerning weakly compact operators or [3] and [6] concerning unconditionally converging operators) were given under additional assumptions on the Banach space E. The aim of this paper is to show that one cannot drop the assumptions on E, indeed as we shall soon show many of the operator characterizations characterize the Banach space E itself. More specifically, it is known [3] that if E* and E** have the Radon-Nikodym property then a bounded linear operator T: C(X, E) → F is weakly compact if and only if the measure G is continuous at Ø (also called strongly bounded), i.e. limn ||G|| (Bn) = 0 for every decreasing sequence Bn ↘ Ø of Borel subsets of X (here ||G|| (B) denotes the semivariation of G at B), and if for every Borel set B the operator G(B) is a weakly compact operator from E to F. In this paper we shall show that if one wants to characterize weakly compact operators as those operators with the above mentioned properties then E* and E** must both have the Radon-Nikodym property. This will constitute the first part of this paper and answers in the negative a question of [2]. In the second part we consider unconditionally converging operators on C(X, E). It is known [6] that if T: C(X, E) → F is an unconditionally converging operator, then its representing measure G is continuous at 0 and, for every Borel set B, G(B) is an unconditionally converging operator from E to F. The converse of the above result was shown to be untrue by a nice example (see [2]). Here again we show that if one wants to characterize unconditionally converging operators as above, then the Banach space E cannot contain a copy of c0. Finally, in the last section we characterize Banach spaces E with the Schur property in terms of properties of Dunford-Pettis operators on C(X, E) spaces.

scholarly journals Double-dual types over the Banach spaceC(K)

2005 ◽  
Vol 2005 (16) ◽  
pp. 2533-2545
Author(s):  
Markus Pomper
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Lower Semicontinuous ◽  
Continuous Functions ◽  
Upper Semicontinuous ◽  
Isolated Points

LetKbe a compact Hausdorff space andC(K)the Banach space of all real-valued continuous functions onK, with the sup-norm. Types overC(K)(in the sense of Krivine and Maurey) can be uniquely represented by pairs(ℓ,u)of bounded real-valued functions onK, whereℓis lower semicontinuous,uis upper semicontinuous,ℓ≤u, andℓ(x)=u(x)for all isolated pointsxofK. A condition that characterizes the pairs(ℓ,u)that represent double-dual types overC(K)is given.

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