the bartle dunford schwartz and the dinculeanu singer ... · extension of the dinculeanu{singer...

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

[email protected], [email protected], [email protected]

5th Workshop on Functional AnalysisValencia, 17–20 October 2017

F. Munoz, E. Oja, C. Pineiro Classical Representation Theorems Revisited 1/18

F. Munoz, E. Oja, C. Pineiro,

The Bartle–Dunford–Schwartz and the Dinculeanu–Singertheorems revisited,

arXiv:1612.07312 [math.FA] (2016) 1–27.

1 Introduction

2 Extension of the Bartle–Dunford–Schwartz Theorem

3 Extension of the Dinculeanu–Singer Theorem

4 q-Semivariation

1 Introduction

4 q-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

∫Ωϕ dµ for all ϕ ∈ C(Ω).

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.

(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

∫Ωf dm for all f ∈ C(Ω,X ).

(D–S) U ∈ L(C(Ω,X ),Y ) ↔ m : Σ→ L(X ,Y ∗∗)

(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

∫Ωf dm for all f ∈ C(Ω,X ).

(D–S) U ∈ L(C(Ω,X ),Y ) ↔ m : Σ→ L(X ,Y ∗∗)

1 Introduction

4 q-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 ∗.

(B–D–S) S ∈ L(C(Ω),Y ) ↔ µ : Σ→ Y ∗∗

Sϕ =∫

Sxϕ = (Sϕ)x , ϕ ∈ C(Ω).

(B–D–S) Sx ∈ L(C(Ω),Y ) ↔ mx : Σ→ Y ∗∗

(B–D–S) S ∈ L(C(Ω),Y ) ↔ µ : Σ→ Y ∗∗

Sϕ =∫

Sxϕ = (Sϕ)x , ϕ ∈ C(Ω).

(B–D–S) Sx ∈ L(C(Ω),Y ) ↔ mx : Σ→ Y ∗∗

(B–D–S) S ∈ L(C(Ω),Y ) ↔ µ : Σ→ Y ∗∗

Sϕ =∫

Sxϕ = (Sϕ)x , ϕ ∈ C(Ω).

(B–D–S) Sx ∈ L(C(Ω),Y ) ↔ mx : Σ→ Y ∗∗

(B–D–S) S ∈ L(C(Ω),Y ) ↔ µ : Σ→ Y ∗∗

Sϕ =∫

Sxϕ = (Sϕ)x , ϕ ∈ C(Ω).

(B–D–S) Sx ∈ L(C(Ω),Y ) ↔ mx : Σ→ Y ∗∗

〈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

(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 ∗∗)

(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 ∗∗)

Question.

What is the relation between µ : Σ→ L(X ,Y )∗∗ andm : Σ→ L(X ,Y ∗∗)?

In general, they are different. Let

J : L(X ,Y )→ L(X ,Y ∗∗), J(A) = jYA, A ∈ L(X ,Y ),

be the natural isometric embedding, where jY : Y → Y ∗∗, and

P : L(X ,Y ∗∗)∗∗ → L(X ,Y ∗∗)

be the natural projection.

Corollary

m(E ) = PJ∗∗µ(E ) for all E ∈ Σ.Moreover, if S is weakly compact, then m = µ : Σ→ L(X ,Y ).

Question.

In general, they are different.

P : L(X ,Y ∗∗)∗∗ → L(X ,Y ∗∗)

Corollary

Question.

P : L(X ,Y ∗∗)∗∗ → L(X ,Y ∗∗)

Corollary

Question.

P : L(X ,Y ∗∗)∗∗ → L(X ,Y ∗∗)

Corollary

1 Introduction

4 q-Semivariation

(D–S) U ∈ L(C(Ω,X ),Y ) ↔ m : Σ→ L(X ,Y ∗∗)

Theorem (Grothendieck compactness principle, 1955).

K ⊂ X is relatively compact iff there exists (xn) ∈ c0(X ) suchthat

K ⊂ ∑

n anxn : (an) ∈ B`1.

Definition (Sinha–Karn, 2002).

Let 1 ≤ p ≤ ∞ and 1/p + 1/p′ = 1 (with 1/∞ = 0).K ⊂ X is relatively p-compact iff there exists (xn) ∈ `p(X )((xn) ∈ c0(X ) if p =∞) such that

K ⊂ p-co(xn) := ∑

n anxn : (an) ∈ B`p′.

p =∞ ⇒ ∞-compact set = compact set.

1 ≤ p ≤ q ≤ ∞ ⇒ p-compact sets are q-compact.Of course, p-compact sets are compact.

(D–S) U ∈ L(C(Ω,X ),Y ) ↔ m : Σ→ L(X ,Y ∗∗)

K ⊂ ∑

K ⊂ p-co(xn) := ∑

(D–S) U ∈ L(C(Ω,X ),Y ) ↔ m : Σ→ L(X ,Y ∗∗)

K ⊂ ∑

K ⊂ p-co(xn) := ∑

(D–S) U ∈ L(C(Ω,X ),Y ) ↔ m : Σ→ L(X ,Y ∗∗)

K ⊂ ∑

K ⊂ p-co(xn) := ∑

(D–S) U ∈ L(C(Ω,X ),Y ) ↔ m : Σ→ L(X ,Y ∗∗)

K ⊂ ∑

K ⊂ p-co(xn) := ∑

Definition: p-continuous X -valued functions(Munoz–Oja–Pineiro, 2015).

Let X be a Banach space and let Ω be a compact Hausdorff space.Let 1 ≤ p ≤ ∞.

Cp(Ω,X ) = f ∈ C(Ω,X ) : f (Ω) is p-compact

C∞(Ω,X ) = C(Ω,X ) as sets.

Theorem (Munoz–Oja–Pineiro, 2015).

Cp(Ω,X ) = C(Ω)⊗dpX ,

where dp represents the right Chevet-Saphar tensor norm.

C(Ω)⊗d∞X = C(Ω)⊗εX = C(Ω,X )

↑ Saphar (1970) ↑ Grothendieck (1955)

C(Ω)⊗d∞X = C(Ω)⊗εX = C(Ω,X )

C(Ω)⊗d∞X

= C(Ω)⊗εX = C(Ω,X )

C(Ω)⊗d∞X = C(Ω)⊗εX

= C(Ω,X )

↑ Saphar (1970)

↑ Grothendieck (1955)

C(Ω)⊗d∞X = C(Ω)⊗εX = C(Ω,X )

↑ Saphar (1970) ↑ Grothendieck (1955)F. Munoz, E. Oja, C. Pineiro Classical Representation Theorems Revisited 13/18

(D–S) U ∈ L(C(Ω,X ),Y ) ↔ m : Σ→ L(X ,Y ∗∗)

Theorem (Extension of Dinculeanu–Singer Th.)

Let 1 ≤ p ≤ ∞. For every operator U ∈ L(Cp(Ω,X ),Y ) thereexists a unique vector measure m : Σ→ L(X ,Y ∗∗) of boundedp′-semivariation such that

∫Ωf dm for all f ∈ Cp(Ω,X ).

(Ext. B–D–S) S ∈ L(C(Ω),L(X ,Y )) ↔ m : Σ→ L(X ,Y ∗∗)(Ext. D–S) U ∈ L(Cp(Ω,X ),Y ) ↔ m : Σ→ L(X ,Y ∗∗)

(D–S) U ∈ L(C(Ω,X ),Y ) ↔ m : Σ→ L(X ,Y ∗∗)

Given U ∈ L(Cp(Ω,X ),Y ), we consider the associatedoperator U# ∈ L(C(Ω),L(X ,Y )), defined by

(U#ϕ)x = U(ϕx) for all ϕ ∈ C(Ω), x ∈ X .

Notice that U# is of the type of S!!!.

By (Ext. B–D–S),U# ∈ L(C(Ω),L(X ,Y )) ↔ m : Σ→ L(X ,Y ∗∗)

Then, we show thatU ∈ L(Cp(Ω,X ),Y ) ↔ m : Σ→ L(X ,Y ∗∗)

Therefore, we have thatRepresenting measure of U# ⇒ Representing measure of U

One easily shows thatRepresenting measure of U# ⇐ Representing measure of U

Thus, both measures coincide.

1 Introduction

4 q-Semivariation

S ∈ L(C(Ω),L(X ,Y )) U ∈ L(Cp(Ω,X ),Y ) U ∈ L(C(Ω,X ),Y )

m : Σ→ L(X ,Y ∗∗)∫Ωϕ dm

∫Ωf dm

‖m‖(Ω) <∞ ‖m‖p′(Ω) <∞ ‖m‖(Ω) <∞

‖m‖(Ω) = supy∗∈BY∗

|my∗ |(Ω), with my∗ := (m(·))∗y∗ : Σ→ X ∗

For every y∗ ∈ Y ∗, let my∗ := (m(·))∗y∗ : Σ→ X ∗.Consider the integral operator Iy∗ ∈ L(C(Ω),X ∗) defined by

Iy∗ϕ =∫

Ω ϕ dmy∗ .

Definition (q-semivariation).

1 ≤ q ≤ ∞‖m‖q(Ω) = supy∗∈BY∗ ‖Iy∗‖Pq

m : Σ→ L(X ,Y ∗∗)

∫Ωϕ dm

∫Ωf dm

‖m‖(Ω) <∞ ‖m‖p′(Ω) <∞ ‖m‖(Ω) <∞

|my∗ |(Ω), with my∗ := (m(·))∗y∗ : Σ→ X ∗

Iy∗ϕ =∫

Ω ϕ dmy∗ .

1 ≤ q ≤ ∞‖m‖q(Ω) = supy∗∈BY∗ ‖Iy∗‖Pq

m : Σ→ L(X ,Y ∗∗)∫Ωϕ dm

∫Ωf dm

‖m‖(Ω) <∞

‖m‖p′(Ω) <∞ ‖m‖(Ω) <∞

‖m‖(Ω) = sup∥∥∥∑Ei∈Π εim(Ei )

∥∥∥Π = (Ei )

ni=1 finite partitions of Ω and all finite systems (εi )

with |εi | ≤ 1, 1 ≤ i ≤ n

|my∗ |(Ω), with my∗ := (m(·))∗y∗ : Σ→ X ∗

Iy∗ϕ =∫

Ω ϕ dmy∗ .

1 ≤ q ≤ ∞‖m‖q(Ω) = supy∗∈BY∗ ‖Iy∗‖Pq

m : Σ→ L(X ,Y ∗∗)∫Ωϕ dm

∫Ωf dm

‖m‖(Ω) <∞

‖m‖p′(Ω) <∞

‖m‖G−D(Ω) <∞

‖m‖G−D(Ω) = sup∥∥∥∑Ei∈Π m(Ei )xi

∥∥∥Π = (Ei )

ni=1 finite partitions of Ω and all finite systems (xi )

with ‖xi‖ ≤ 1

|my∗ |(Ω), with my∗ := (m(·))∗y∗ : Σ→ X ∗

Iy∗ϕ =∫

Ω ϕ dmy∗ .

1 ≤ q ≤ ∞‖m‖q(Ω) = supy∗∈BY∗ ‖Iy∗‖Pq

m : Σ→ L(X ,Y ∗∗)∫Ωϕ dm

∫Ωf dm

‖m‖(Ω) <∞ ‖m‖p′(Ω) <∞ ‖m‖G−D(Ω) <∞

|my∗ |(Ω), with my∗ := (m(·))∗y∗ : Σ→ X ∗

Iy∗ϕ =∫

Ω ϕ dmy∗ .

1 ≤ q ≤ ∞‖m‖q(Ω) = supy∗∈BY∗ ‖Iy∗‖Pq

m : Σ→ L(X ,Y ∗∗)∫Ωϕ dm

∫Ωf dm

‖m‖(Ω) <∞ ‖m‖p′(Ω) <∞ ‖m‖G−D(Ω) <∞

‖m‖G−D(Ω) = supy∗∈BY∗

|my∗ |(Ω), with my∗ := (m(·))∗y∗ : Σ→ X ∗

Iy∗ϕ =∫

Ω ϕ dmy∗ .

1 ≤ q ≤ ∞‖m‖q(Ω) = supy∗∈BY∗ ‖Iy∗‖Pq

m : Σ→ L(X ,Y ∗∗)∫Ωϕ dm

∫Ωf dm

‖m‖(Ω) <∞ ‖m‖p′(Ω) <∞ ‖m‖G−D(Ω) <∞

|my∗ |(Ω), with my∗ := (m(·))∗y∗ : Σ→ X ∗

Iy∗ϕ =∫

Ω ϕ dmy∗ .

1 ≤ q ≤ ∞‖m‖q(Ω) = supy∗∈BY∗ ‖Iy∗‖Pq

m : Σ→ L(X ,Y ∗∗)∫Ωϕ dm

∫Ωf dm

‖m‖(Ω) <∞ ‖m‖p′(Ω) <∞ ‖m‖G−D(Ω) <∞

|my∗ |(Ω), with my∗ := (m(·))∗y∗ : Σ→ X ∗

Iy∗ϕ =∫

Ω ϕ dmy∗ .

1 ≤ q ≤ ∞‖m‖q(Ω) = supy∗∈BY∗ ‖Iy∗‖Pq

m : Σ→ L(X ,Y ∗∗)∫Ωϕ dm

∫Ωf dm

‖m‖∞(Ω) <∞ ‖m‖p′(Ω) <∞ ‖m‖G−D(Ω) <∞

|my∗ |(Ω), with my∗ := (m(·))∗y∗ : Σ→ X ∗

Iy∗ϕ =∫

Ω ϕ dmy∗ .

1 ≤ q ≤ ∞‖m‖q(Ω) = supy∗∈BY∗ ‖Iy∗‖Pq

m : Σ→ L(X ,Y ∗∗)∫Ωϕ dm

∫Ωf dm

‖m‖∞(Ω) <∞ ‖m‖p′(Ω) <∞ ‖m‖1(Ω) <∞

‖m‖1(Ω) = supy∗∈BY∗

|my∗ |(Ω), with my∗ := (m(·))∗y∗ : Σ→ X ∗

Iy∗ϕ =∫

Ω ϕ dmy∗ .

1 ≤ q ≤ ∞‖m‖q(Ω) = supy∗∈BY∗ ‖Iy∗‖Pq

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

[email protected], [email protected], [email protected]

5th Workshop on Functional AnalysisValencia, 17–20 October 2017

the bartle dunford schwartz and the dinculeanu singer ... · extension of the dinculeanu{singer...

Documents