scholarly journals Another proof of a result of N. J. Kalton, E. Saab and P. Saab on the Dieudonné property in C(K, E)

1989 ◽  
Vol 31 (2) ◽  
pp. 137-140 ◽  
Author(s):  
G. Emmanuele
Keyword(s):  
Vector Measure ◽  
Measure Theory ◽  
Short Note ◽  
Continuous Operator ◽  
Weak Compactness ◽  
Multivalued Mappings ◽  
Weakly Compact ◽  
Completely Continuous ◽  
Completely Continuous Operator ◽  
Integrable Functions

Let K be a compact Hausdorff topological space and E be a Banach space not containing l1. Recently N. J. Kalton, E. Saab and P. Saab ([5]) obtained the results that under the above assumptions the usual space C(K, E) has the Dieudonné property; i.e. each weakly completely continuous operator on C(K, E) is weakly compact. They use topological results concerning multivalued mappings in their proof. In this short note we furnish a new and simpler proof of that result without using topological results but only well known theorems of Bourgain ([2]) and Talagrand ([8]) on weak compactness of sets of Bochner integrable functions; i.e. results in vector measure theory. At the end of the paper we present some applications of the result to Banach spaces of compact operators.

scholarly journals Sufficient conditions for a continuous linear operator to be weakly compact

1972 ◽  
Vol 7 (2) ◽  
pp. 183-190 ◽  
Author(s):  
Joe Howard ◽  
Kenneth Melendez
Keyword(s):  
Linear Operator ◽  
Sufficient Conditions ◽  
Continuous Linear Operator ◽  
Continuous Operator ◽  
Continuous Linear ◽  
Weakly Compact ◽  
Completely Continuous ◽  
Completely Continuous Operator ◽  
Locally Convex

A locally convex topological vector (LCTV) space E is said to have property V (Dieudonné property) if for every complete separated LCTV space F, every unconditionally converging (weakly completely continuous) operator T: E → F is wsakly compact. First, an investigation of the permanence of property V is given. The permanence of the Dieudonné is analogous. Relationships between property V and the Dieudonné property are then given.

scholarly journals Non-weak compactness of the integration map for vector measures

1993 ◽  
Vol 54 (3) ◽  
pp. 287-303 ◽  
Author(s):  
S. Okada ◽  
W. J. Ricker
Keyword(s):  
Banach Space ◽  
Volterra Operator ◽  
Continuous Operator ◽  
Weakly Compact ◽  
Vector Measures ◽  
Mean Convergence ◽  
Integrable Functions ◽  
The Mean ◽  
The Fourier Transform ◽  
Adjoint Operators

AbstractLet m be a vector measure with values in a Banach space X. If L1(m) denotes the space of all m integrable functions then, with respect to the mean convergence topology, L1(m) is a Banach space. A natural operator associated with m is its integration map Im which sends each f of L1(m) to the element ∫fdm (of X). Many properties of the (continuous) operator Im are closely related to the nature of the space L1(m). In general, it is difficult to identify L1(m). We aim to exhibit non-trivial examples of measures m in (non-reflexive) spaces X for which L1(m) can be explicitly computed and such that Im is not weakly compact. The examples include some well known operators from analysis (the Fourier transform on L1 ([−π, π]), the Volterra operator on L1 ([0, 1]), compact self-adjoint operators in a Hilbert space); such operators can be identified with integration maps Im (or their restrictions) for suitable measures m.

The Uncomplemented Subspace K(X,Y)

2013 ◽  
Vol 56 (1) ◽  
pp. 65-69
Author(s):  
Ioana Ghenciu
Keyword(s):  
Vector Measure ◽  
Compact Operators ◽  
Weakly Compact ◽  
Completely Continuous ◽  
Spaces Of Operators ◽  
Completely Continuous Operators ◽  
Weakly Compact Operators ◽  
Continuous Operators

AbstractA vector measure result is used to study the complementation of the space K(X,Y) of compact operators in the spaces W(X,Y) of weakly compact operators, CC(X,Y) of completely continuous operators, and U(X,Y) of unconditionally converging operators. Results of Kalton and Emmanuele concerning the complementation of K(X,Y) in L(X,Y) and in W(X,Y) are generalized. The containment of c0 and ℓ∞ in spaces of operators is also studied.

scholarly journals Remarks on Weak Compactness in L1(μ,X)

1977 ◽  
Vol 18 (1) ◽  
pp. 87-91 ◽  
Author(s):  
J. Diestel
Keyword(s):  
Banach Space ◽  
Sufficient Conditions ◽  
Finite Measure ◽  
Factorization Method ◽  
Weak Compactness ◽  
Weakly Compact ◽  
Integrable Functions ◽  
Nikodym Property ◽  
Potential Applicability ◽  
Image Position

Let (Ω,Σ,μ) be a finite measure space and X a Banach space. Denote by L1 (μ,X) the Banach space of (equivalence classes of) μ-strongly measurable X-valued Bochner integrable functions f:Ω→X normed byThe problem of characterizing the relatively weakly compact subsets of L1(Ω, X) remains open. It is known that for a bounded subset of L1(μ, X) to be relatively weakly compact it is necessary that the set be uniformly integrable; recall that K ⊆ L1, (μ, X) is uniformly integrable whenever given ε >0 there exists δ > 0 such that if μ (E) ≦ δ then ∫E∥f∥ dμ ≦ δ, for all f ∈ K. S. Chatterji has noted that in case X is reflexive this condition is also sufficient [4]. At present unless one assumes that both X and X* have the Radon-Nikodym Property (see [1]), a rather severe restriction which, for purposes of potential applicability, is tantamount to assuming reflexivity, no good sufficient conditions for weak compactness in L1(μ, X) exist. This note puts forth such sufficient conditions; the basic tool is the recent factorization method of W. J. Davis, T. Figiel, W. B. Johnson and A. Pelczynski [3].

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