Introduction Extension of the Bartle–Dunford–Schwartz Theorem Extension of the Dinculeanu–Singer Theorem q-Semivariation The Bartle–Dunford–Schwartz and the Dinculeanu–Singer Theorems Revisited Fernando Mu˜ noz 1 Eve Oja 2 C´ andido Pi˜ neiro 3 1,3 University of Huelva (Spain) Department of Integrated Science 2 University of Tartu (Estonia) Institute of Mathematics and Statistics 1 [email protected], 2 [email protected], 3 [email protected]5th Workshop on Functional Analysis Valencia, 17–20 October 2017 F. Mu˜ noz, E. Oja, C. Pi˜ neiro Classical Representation Theorems Revisited 1/18
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IntroductionExtension of the Bartle–Dunford–Schwartz Theorem
Extension of the Dinculeanu–Singer Theoremq-Semivariation
The Bartle–Dunford–Schwartz and theDinculeanu–Singer Theorems Revisited
Fernando Munoz1 Eve Oja2 Candido Pineiro3
1,3 University of Huelva (Spain)Department of Integrated Science
2 University of Tartu (Estonia)Institute of Mathematics and Statistics
5th Workshop on Functional AnalysisValencia, 17–20 October 2017
F. Munoz, E. Oja, C. Pineiro Classical Representation Theorems Revisited 1/18
IntroductionExtension of the Bartle–Dunford–Schwartz Theorem
Extension of the Dinculeanu–Singer Theoremq-Semivariation
F. Munoz, E. Oja, C. Pineiro,
The Bartle–Dunford–Schwartz and the Dinculeanu–Singertheorems revisited,
arXiv:1612.07312 [math.FA] (2016) 1–27.
F. Munoz, E. Oja, C. Pineiro Classical Representation Theorems Revisited 2/18
IntroductionExtension of the Bartle–Dunford–Schwartz Theorem
Extension of the Dinculeanu–Singer Theoremq-Semivariation
Index
1 Introduction
2 Extension of the Bartle–Dunford–Schwartz Theorem
3 Extension of the Dinculeanu–Singer Theorem
4 q-Semivariation
F. Munoz, E. Oja, C. Pineiro Classical Representation Theorems Revisited 3/18
IntroductionExtension of the Bartle–Dunford–Schwartz Theorem
Extension of the Dinculeanu–Singer Theoremq-Semivariation
Index
1 Introduction
2 Extension of the Bartle–Dunford–Schwartz Theorem
3 Extension of the Dinculeanu–Singer Theorem
4 q-Semivariation
F. Munoz, E. Oja, C. Pineiro Classical Representation Theorems Revisited 4/18
IntroductionExtension of the Bartle–Dunford–Schwartz Theorem
Extension of the Dinculeanu–Singer Theoremq-Semivariation
Theorem (Bartle–Dunford–Schwartz, 1955).
For every operator S ∈ L(C(Ω),Y ) there exists a unique vectormeasure µ : Σ→ Y ∗∗ of bounded semivariation such that
Sϕ =
∫Ωϕ dµ for all ϕ ∈ C(Ω).
F. Munoz, E. Oja, C. Pineiro Classical Representation Theorems Revisited 5/18
IntroductionExtension of the Bartle–Dunford–Schwartz Theorem
Extension of the Dinculeanu–Singer Theoremq-Semivariation
Theorem (Bartle–Dunford–Schwartz, 1955).
Let Y be a Banach space and let Ω be a compact Hausdorff space.For every S ∈ L(C(Ω),Y ) there exists a weak*-countably additivemeasure µ : Σ→ Y ∗∗ such that(i) 〈µ(·), y∗〉 is a regular countably additive Borel measure
for each y∗ ∈ Y ∗;(ii) the map Y ∗ → C(Ω)∗, y∗ 7→ 〈µ(·), y∗〉, is weak*-to-weak*
continuous;(iii) 〈Sϕ, y∗〉 =
∫Ω ϕ d(〈µ(·), y∗〉), for each ϕ ∈ C(Ω) and
each y∗ ∈ Y ∗; and(iv) ‖S‖ = ‖µ‖(Ω).
Conversely, any vector measure µ : Σ→ Y ∗∗ that satisfies (i) and(ii) defines an operator S ∈ L(C(Ω),Y ) by means of (iii), and (iv)follows.
F. Munoz, E. Oja, C. Pineiro Classical Representation Theorems Revisited 6/18
IntroductionExtension of the Bartle–Dunford–Schwartz Theorem
Extension of the Dinculeanu–Singer Theoremq-Semivariation
Theorem (Bartle–Dunford–Schwartz, 1955).
For every operator S ∈ L(C(Ω),Y ) there exists a unique vectormeasure µ : Σ→ Y ∗∗ of bounded semivariation such that
Sϕ =
∫Ωϕ dµ for all ϕ ∈ C(Ω).
(B–D–S) S ∈ L(C(Ω),Y ) ↔ µ : Σ→ Y ∗∗
Theorem (Dinculeanu–Singer, 1959, 1965).
For every operator U ∈ L(C(Ω,X ),Y ) there exists a unique vectormeasure m : Σ→ L(X ,Y ∗∗) of bounded 1-semivariation such that
Uf =
∫Ωf dm for all f ∈ C(Ω,X ).
(D–S) U ∈ L(C(Ω,X ),Y ) ↔ m : Σ→ L(X ,Y ∗∗)
F. Munoz, E. Oja, C. Pineiro Classical Representation Theorems Revisited 6/18
IntroductionExtension of the Bartle–Dunford–Schwartz Theorem
Extension of the Dinculeanu–Singer Theoremq-Semivariation
Theorem (Bartle–Dunford–Schwartz, 1955).
For every operator S ∈ L(C(Ω),Y ) there exists a unique vectormeasure µ : Σ→ Y ∗∗ of bounded semivariation such that
Sϕ =
∫Ωϕ dµ for all ϕ ∈ C(Ω).
(B–D–S) S ∈ L(C(Ω),Y ) ↔ µ : Σ→ Y ∗∗
Theorem (Dinculeanu–Singer, 1959, 1965).
For every operator U ∈ L(C(Ω,X ),Y ) there exists a unique vectormeasure m : Σ→ L(X ,Y ∗∗) of bounded 1-semivariation such that
Uf =
∫Ωf dm for all f ∈ C(Ω,X ).
(D–S) U ∈ L(C(Ω,X ),Y ) ↔ m : Σ→ L(X ,Y ∗∗)
F. Munoz, E. Oja, C. Pineiro Classical Representation Theorems Revisited 6/18
IntroductionExtension of the Bartle–Dunford–Schwartz Theorem
Extension of the Dinculeanu–Singer Theoremq-Semivariation
Index
1 Introduction
2 Extension of the Bartle–Dunford–Schwartz Theorem
3 Extension of the Dinculeanu–Singer Theorem
4 q-Semivariation
F. Munoz, E. Oja, C. Pineiro Classical Representation Theorems Revisited 7/18
IntroductionExtension of the Bartle–Dunford–Schwartz Theorem
Extension of the Dinculeanu–Singer Theoremq-Semivariation
(B–D–S) S ∈ L(C(Ω),Y ) ↔ µ : Σ→ Y ∗∗
If we replace Y = L(K,Y ) by L(X ,Y ), we obtain
(B–D–S) S ∈ L(C(Ω),L(X ,Y )) ↔ µ : Σ→ L(X ,Y )∗∗
Theorem (Extension of Bartle–Dunford–Schwartz Th.)
For every operator S ∈ L(C(Ω),L(X ,Y )) there exists a uniquevector measure m : Σ→ L(X ,Y ∗∗) of bounded semivariation suchthat
Sϕ =∫
Ω ϕ dm for all ϕ ∈ C(Ω).
For every x ∈ X , we define an operator Sx ∈ L(C(Ω),Y ) by
Sxϕ = (Sϕ)x , ϕ ∈ C(Ω).
(B–D–S) Sx ∈ L(C(Ω),Y ) ↔ mx : Σ→ Y ∗∗
We define m : Σ→ L(X ,Y ∗∗) by
〈y∗,m(E )x〉 = 〈y∗,mx(E )〉, for all x ∈ X and y∗ ∈ Y ∗.
F. Munoz, E. Oja, C. Pineiro Classical Representation Theorems Revisited 8/18
IntroductionExtension of the Bartle–Dunford–Schwartz Theorem
Extension of the Dinculeanu–Singer Theoremq-Semivariation
(B–D–S) S ∈ L(C(Ω),Y ) ↔ µ : Σ→ Y ∗∗
If we replace Y = L(K,Y ) by L(X ,Y ), we obtain
(B–D–S) S ∈ L(C(Ω),L(X ,Y )) ↔ µ : Σ→ L(X ,Y )∗∗
Theorem (Extension of Bartle–Dunford–Schwartz Th.)
For every operator S ∈ L(C(Ω),L(X ,Y )) there exists a uniquevector measure m : Σ→ L(X ,Y ∗∗) of bounded semivariation suchthat
Sϕ =∫
Ω ϕ dm for all ϕ ∈ C(Ω).
For every x ∈ X , we define an operator Sx ∈ L(C(Ω),Y ) by
Sxϕ = (Sϕ)x , ϕ ∈ C(Ω).
(B–D–S) Sx ∈ L(C(Ω),Y ) ↔ mx : Σ→ Y ∗∗
We define m : Σ→ L(X ,Y ∗∗) by
〈y∗,m(E )x〉 = 〈y∗,mx(E )〉, for all x ∈ X and y∗ ∈ Y ∗.
F. Munoz, E. Oja, C. Pineiro Classical Representation Theorems Revisited 8/18
IntroductionExtension of the Bartle–Dunford–Schwartz Theorem
Extension of the Dinculeanu–Singer Theoremq-Semivariation
(B–D–S) S ∈ L(C(Ω),Y ) ↔ µ : Σ→ Y ∗∗
If we replace Y = L(K,Y ) by L(X ,Y ), we obtain
(B–D–S) S ∈ L(C(Ω),L(X ,Y )) ↔ µ : Σ→ L(X ,Y )∗∗
Theorem (Extension of Bartle–Dunford–Schwartz Th.)
For every operator S ∈ L(C(Ω),L(X ,Y )) there exists a uniquevector measure m : Σ→ L(X ,Y ∗∗) of bounded semivariation suchthat
Sϕ =∫
Ω ϕ dm for all ϕ ∈ C(Ω).
For every x ∈ X , we define an operator Sx ∈ L(C(Ω),Y ) by
Sxϕ = (Sϕ)x , ϕ ∈ C(Ω).
(B–D–S) Sx ∈ L(C(Ω),Y ) ↔ mx : Σ→ Y ∗∗
We define m : Σ→ L(X ,Y ∗∗) by
〈y∗,m(E )x〉 = 〈y∗,mx(E )〉, for all x ∈ X and y∗ ∈ Y ∗.
F. Munoz, E. Oja, C. Pineiro Classical Representation Theorems Revisited 8/18
IntroductionExtension of the Bartle–Dunford–Schwartz Theorem
Extension of the Dinculeanu–Singer Theoremq-Semivariation
(B–D–S) S ∈ L(C(Ω),Y ) ↔ µ : Σ→ Y ∗∗
If we replace Y = L(K,Y ) by L(X ,Y ), we obtain
(B–D–S) S ∈ L(C(Ω),L(X ,Y )) ↔ µ : Σ→ L(X ,Y )∗∗
Theorem (Extension of Bartle–Dunford–Schwartz Th.)
For every operator S ∈ L(C(Ω),L(X ,Y )) there exists a uniquevector measure m : Σ→ L(X ,Y ∗∗) of bounded semivariation suchthat
Sϕ =∫
Ω ϕ dm for all ϕ ∈ C(Ω).
For every x ∈ X , we define an operator Sx ∈ L(C(Ω),Y ) by
Sxϕ = (Sϕ)x , ϕ ∈ C(Ω).
(B–D–S) Sx ∈ L(C(Ω),Y ) ↔ mx : Σ→ Y ∗∗
We define m : Σ→ L(X ,Y ∗∗) by
〈y∗,m(E )x〉 = 〈y∗,mx(E )〉, for all x ∈ X and y∗ ∈ Y ∗.
F. Munoz, E. Oja, C. Pineiro Classical Representation Theorems Revisited 8/18
IntroductionExtension of the Bartle–Dunford–Schwartz Theorem
Extension of the Dinculeanu–Singer Theoremq-Semivariation
(B–D–S) S ∈ L(C(Ω),Y ) ↔ µ : Σ→ Y ∗∗
If we replace Y = L(K,Y ) by L(X ,Y ), we obtain
(B–D–S) S ∈ L(C(Ω),L(X ,Y )) ↔ µ : Σ→ L(X ,Y )∗∗
Theorem (Extension of Bartle–Dunford–Schwartz Th.)
For every operator S ∈ L(C(Ω),L(X ,Y )) there exists a uniquevector measure m : Σ→ L(X ,Y ∗∗) of bounded semivariation suchthat
Sϕ =∫
Ω ϕ dm for all ϕ ∈ C(Ω).
For every x ∈ X , we define an operator Sx ∈ L(C(Ω),Y ) by
Sxϕ = (Sϕ)x , ϕ ∈ C(Ω).
(B–D–S) Sx ∈ L(C(Ω),Y ) ↔ mx : Σ→ Y ∗∗
We define m : Σ→ L(X ,Y ∗∗) by
〈y∗,m(E )x〉 = 〈y∗,mx(E )〉, for all x ∈ X and y∗ ∈ Y ∗.F. Munoz, E. Oja, C. Pineiro Classical Representation Theorems Revisited 8/18
IntroductionExtension of the Bartle–Dunford–Schwartz Theorem
Extension of the Dinculeanu–Singer Theoremq-Semivariation
(B–D–S) S ∈ L(C(Ω),L(X ,Y )) ↔ µ : Σ→ L(X ,Y )∗∗
(Ext. B–D–S) S ∈ L(C(Ω),L(X ,Y )) ↔ m : Σ→ L(X ,Y ∗∗)
(D–S) U ∈ L(C(Ω,X ),Y ) ↔ m : Σ→ L(X ,Y ∗∗)
F. Munoz, E. Oja, C. Pineiro Classical Representation Theorems Revisited 9/18
IntroductionExtension of the Bartle–Dunford–Schwartz Theorem
Extension of the Dinculeanu–Singer Theoremq-Semivariation
(B–D–S) S ∈ L(C(Ω),L(X ,Y )) ↔ µ : Σ→ L(X ,Y )∗∗
(Ext. B–D–S) S ∈ L(C(Ω),L(X ,Y )) ↔ m : Σ→ L(X ,Y ∗∗)(D–S) U ∈ L(C(Ω,X ),Y ) ↔ m : Σ→ L(X ,Y ∗∗)
F. Munoz, E. Oja, C. Pineiro Classical Representation Theorems Revisited 9/18
IntroductionExtension of the Bartle–Dunford–Schwartz Theorem
Extension of the Dinculeanu–Singer Theoremq-Semivariation
Question.
What is the relation between µ : Σ→ L(X ,Y )∗∗ andm : Σ→ L(X ,Y ∗∗)?