scholarly journals Nuclear operators on spaces of continuous vector-valued functions

1991 ◽  
Vol 33 (2) ◽  
pp. 223-230 ◽  
Author(s):  
Paulette Saab ◽  
Brenda Smith
Keyword(s):  
Banach Space ◽  
Vector Measure ◽  
Type Theorem ◽  
Representing Measure ◽  
Bounded Linear ◽  
Continuous Vector ◽  
Nuclear Operators ◽  
Nikodym Property ◽  
Vector Valued Functions ◽  
Vector Valued

Let Ω: be a compact Hausdorff space, let E be a Banach space, and let C(Ω, E) stand for the Banach space of continuous E-valued functions on Ω under supnorm. It is well known [3, p. 182] that if F is a Banach space then any bounded linear operator T:C(Ω, E)→ F has a finitely additive vector measure G defined on the σ-field of Borel subsets of Ω with values in the space ℒ(E, F**) of bounded linear operators from E to the second dual F** of F. The measure G is said to represent T. The purpose of this note is to study the interplay between certain properties of the operator T and properties of the representing measure G. Precisely, one of our goals is to study when one can characterize nuclear operators in terms of their representing measures. This is of course motivated by a well-known theorem of L. Schwartz [5] (see also [3, p. 173]) concerning nuclear operators on spaces C(Ω) of continuous scalar-valued functions. The study of nuclear operators on spaces C(Ω, E) of continuous vector-valued functions was initiated in [1], where the author extended Schwartz's result in case E* has the Radon-Nikodym property. In this paper, we will show that the condition on E* to have the Radon-Nikodym property is necessary to have a Schwartz's type theorem. This leads to a new characterization of dual spaces E* with the Radon-Nikodym property. In [2], it was shown that if T:C(Ω, E)→ F is nuclear than its representing measure G takes its values in the space (E, F) of nuclear operators from E to F. One of the results of this paper is that if T:C(Ω, E)→ F is nuclear then its representing measure G is countably additive and of bounded variation as a vector measure taking its values in (E, F) equipped with the nuclear norm. Finally, we show by easy examples that the above mentioned conditions on the representing measure G do not characterize nuclear operators on C(Ω, E) spaces, and we also look at cases where nuclear operators are indeed characterized by the above two conditions. For all undefined notions and terminologies, we refer the reader to [3].

A Choquet theorem for general subspaces of vector-valued functions

1985 ◽  
Vol 98 (2) ◽  
pp. 323-326 ◽  
Author(s):  
Paulette Saab ◽  
Michel Talagrand
Keyword(s):  
Banach Space ◽  
Hausdorff Space ◽  
Linear Subspace ◽  
Vector Measure ◽  
Complex Banach Space ◽  
Supremum Norm ◽  
Bounded Linear ◽  
Vector Valued Functions ◽  
Bounded Linear Functional ◽  
Vector Valued

Let X be a compact Hausdorff space, let E be a (real or complex) Banach space, and let C(X, E) stand for the Banach space of all continuous E-valued functions defined on X under the supremum norm. If A is an arbitrary linear subspace of C(X, E), then it is shown that each bounded linear functional l on A can be represented by a boundary E*-valued vector measure μ on X that has the same norm as l. This result constitutes an extension to vector-valued functions of the so-called analytic version of Choquet's integral representation theorem.

scholarly journals Nuclear operators on Banach function spaces

Positivity ◽  
2020 ◽  
Author(s):  
Marian Nowak
Keyword(s):  
Banach Space ◽  
Convex Space ◽  
Banach Function Space ◽  
Finite Measure ◽  
Locally Convex Space ◽  
Nuclear Operator ◽  
Representing Measure ◽  
Nuclear Operators ◽  
Nikodym Property ◽  
Locally Convex

Abstract Let X be a Banach space and E be a perfect Banach function space over a finite measure space $$(\Omega ,\Sigma ,\lambda )$$ ( Ω , Σ , λ ) such that $$L^\infty \subset E\subset L^1$$ L ∞ ⊂ E ⊂ L 1 . Let $$E'$$ E ′ denote the Köthe dual of E and $$\tau (E,E')$$ τ ( E , E ′ ) stand for the natural Mackey topology on E. It is shown that every nuclear operator $$T:E\rightarrow X$$ T : E → X between the locally convex space $$(E,\tau (E,E'))$$ ( E , τ ( E , E ′ ) ) and a Banach space X is Bochner representable. In particular, we obtain that a linear operator $$T:L^\infty \rightarrow X$$ T : L ∞ → X between the locally convex space $$(L^\infty ,\tau (L^\infty ,L^1))$$ ( L ∞ , τ ( L ∞ , L 1 ) ) and a Banach space X is nuclear if and only if its representing measure $$m_T:\Sigma \rightarrow X$$ m T : Σ → X has the Radon-Nikodym property and $$|m_T|(\Omega )=\Vert T\Vert _{nuc}$$ | m T | ( Ω ) = ‖ T ‖ nuc (= the nuclear norm of T). As an application, it is shown that some natural kernel operators on $$L^\infty $$ L ∞ are nuclear. Moreover, it is shown that every nuclear operator $$T:L^\infty \rightarrow X$$ T : L ∞ → X admits a factorization through some Orlicz space $$L^\varphi $$ L φ , that is, $$T=S\circ i_\infty $$ T = S ∘ i ∞ , where $$S:L^\varphi \rightarrow X$$ S : L φ → X is a Bochner representable and compact operator and $$i_\infty :L^\infty \rightarrow L^\varphi $$ i ∞ : L ∞ → L φ is the inclusion map.

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.

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 POINTWISE CONVERGENCE AND SEMIGROUPS ACTING ON VECTOR-VALUED FUNCTIONS

2011 ◽  
Vol 84 (1) ◽  
pp. 44-48 ◽  
Author(s):  
MICHAEL G. COWLING ◽  
MICHAEL LEINERT
Keyword(s):  
Banach Space ◽  
Pointwise Convergence ◽  
Function Spaces ◽  
Semigroup Of Operators ◽  
Almost Everywhere ◽  
Vector Valued Function ◽  
Vector Valued Functions ◽  
Complex Valued ◽  
Vector Valued

AbstractA submarkovian C0 semigroup (Tt)t∈ℝ+ acting on the scale of complex-valued functions Lp(X,ℂ) extends to a semigroup of operators on the scale of vector-valued function spaces Lp(X,E), when E is a Banach space. It is known that, if f∈Lp(X,ℂ), where 1<p<∞, then Ttf→f pointwise almost everywhere. We show that the same holds when f∈Lp(X,E) .

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