factorization of operators and vector measures

FACTORIZATION OF OPERATORS

AND VECTOR MEASURES

Enrique A. Sanchez Perez

Instituto de Matematica Pura y Aplicada

Universidad Politecnica de Valencia

Factorization of operators and vector measures

An operator on a Banach function space -and also on more generalclasses of Banach spaces of functions- always defines a vector mea-sure. A vector measure is always related in a natural way with aBanach function space -its space of integrable functions-. This isthe starting point of a methodological purpose that deals with vectormeasures, operators and spaces of integrable functions from a uni-fied point of view. It provides information about the structure of theoperators and is often presented in terms of factorization theorems.

There are many old and new results that can be considered as partof this theory. Some lines of research have been developed in this di-rection in the last ten years; for instance, the optimal domain theory,Maurey-Rosenthal type theorems for vector measures on factorizationof operators, the analysis of the topological properties of the integra-tion map -compactness, weak compactness, complete continuity-, andits geometric properties -q-convexity and q-concavity- related to thestructure of the space L1(m).

II Riesz Representation Theorem

II Radon-Nikodym Theorem

VECTOR VALUED VERSIONS:

II Riesz Representation Theorem ⇒ Bartle, Dunford, Schwartz

T : C(K)→ E, T (f) =

∫fdmT .

II Radon-Nikodym Theorem

VECTOR VALUED VERSIONS:

II Riesz Representation Theorem ⇒ Bartle, Dunford, Schwartz

T : C(K)→ E, T (f) =

∫fdmT .

II Radon-Nikodym Theorem ⇒ Diestel, Uhl

T : L1(µ)→ E, T (f) =

∫φfdµ.

L1 of a vector measure

As a Banach space (Lewis, Kluvanek and Knowles, Okada, Ricker, Stefansson)

As a Banach lattice (Curbera)

Vector measures on δ-rings (Masani and Niemi, Delgado)

Extensions and factorization of operators. mT (A) := T (χA)

Optimal domain theorem

X(µ) E-T

HHHj *

L1(mT )i ImT

X(µ) E-T

HHHj *

L1(mT )i ImT

X(µ) E-T

HHHj *

L1(mT )i ImT

Radon-Nikodym derivatives for vector measures

n m ⇔ n(A) =∫A h dm (Musia l)

Where is h? (Calabuig, Gregori, S-P)

L1(m) E-Im

L1(n)i In

Radon-Nikodym derivatives for vector measures

n m ⇔ n(A) =∫A h dm (Musia l)

Where is h? (Calabuig, Gregori, S-P)

L1(m) E-Im

L1(n)i In

II Factorizing operators through spaces of multiplicationoperators (Calabuig, Delgado, S-P. JMAA 2010)

Extensions and semivariation inequalities for operators

1.- Banach function subspaces of L1(n) of a vector measure n on aδ-ring defined by means of semivariations.

2.- Optimal extensions for operators which are bounded by a productnorm.

3.- Examples and Applications.

Subspaces of L1(m) and generalized semivariations

(Ω,Σ, η) measure space. E Banach. Y (η) B.f.s. n : Σ→ E vectormeasure, η n.

Y (η)-semivariation of n

‖n‖Y (η) = sup

∥∥∥∥∫Ωϕdn

∥∥∥∥E

: ϕ ∈ S(Σ) ∩BY (η)

Subspace L1Y (η)(m) =

f ∈ L1(m) : ‖mf‖Y (η) <∞

Norm ‖f‖L1Y (η)

(m) = max ‖f‖m, ‖mf‖Y (η).

Lemma. For every f ∈ L1(m) and ϕ ∈ S(Σ) ∩ Y (η),

‖fϕ‖m ≤ ‖ϕ‖Y (η) ‖mf‖Y (η).

‖n‖Y (η) = sup

∥∥∥∥E

: ϕ ∈ S(Σ) ∩BY (η)

f ∈ L1(m) : ‖mf‖Y (η) <∞

(m) = max ‖f‖m, ‖mf‖Y (η).

‖fϕ‖m ≤ ‖ϕ‖Y (η) ‖mf‖Y (η).

‖n‖Y (η) = sup

∥∥∥∥E

: ϕ ∈ S(Σ) ∩BY (η)

f ∈ L1(m) : ‖mf‖Y (η) <∞

(m) = max ‖f‖m, ‖mf‖Y (η).

‖fϕ‖m ≤ ‖ϕ‖Y (η) ‖mf‖Y (η).

‖n‖Y (η) = sup

∥∥∥∥E

: ϕ ∈ S(Σ) ∩BY (η)

f ∈ L1(m) : ‖mf‖Y (η) <∞

(m) = max ‖f‖m, ‖mf‖Y (η).

‖fϕ‖m ≤ ‖ϕ‖Y (η) ‖mf‖Y (η).

Space of multiplication operators M(Y (η), L1(n)).

Norm: ‖g‖M(Y (η),L1(n)) = supf∈BY (η)‖fg‖L1(n).

It can be written as the dual of a product space (Delgado, S-P, IEOT

2010).

Yb(η) is the closure of simple functions in Y (η).

2010).

Proposition. The equality

L1Y (η)(m) = L1(m) ∩M(Yb(η), L1(m))

holds and ‖mf‖Y (η) = ‖f‖M(Yb(η),L1(m)) for all f ∈ L1Y (η)(m).

L1Y (η)(m) = L1(m) ∩M(Yb(η), L1(m))

Example.

Y (η) = Lp(η) with 1 ≤ p <∞.

L1Lp(η)(m) = L1(m) ∩M(Lp(η), L1(m)) isometrically.

If η is finite L1Lp(η)(m) = M(Lp(η), L1(m)). It coincides with L1

p′,η(m)

( 1p + 1

p′ = 1), given by all functions f ∈ L1(m) such that p′-semivariationof mf with respect to η is finite.

L1Y (η)(m) = L1(m) ∩M(Yb(η), L1(m))

Example.

Y (η) = Lp(η) with 1 ≤ p <∞.

p′,η(m)

( 1p + 1

p′ = 1), given by all functions f ∈ L1(m) such that p′-semivariationof mf with respect to η is finite.

L1Y (η)(m) = L1(m) ∩M(Yb(η), L1(m))

Example.

Y (η) = Lp(η) with 1 ≤ p <∞.

p′,η(m)

( 1p + 1

p′ = 1), given by all functions f ∈ L1(m) such that p′-semivariationof mf with respect to η is finite. That is,

‖mf‖p′,η = supπ∈P(Ω)

supx∗∈BE∗

(∑A∈π

|∫Afdx∗m|p′

η(A)p′−1

)1/p′

(Calabuig, Galaz, Jimenez, S-P, MathZ 2007).

L1Y (η)(m) = L1(m) ∩M(Yb(η), L1(m))

Example.

1 ≤ p <∞, Lp(m) is the 1/p-th power of L1(m), i.e.

Lp(m) = f ∈ L0(|λ|) : |f |p ∈ L1(m).

L1Lp(m)(m) = L1(m) ∩M(Lp(m), L1(m)) with equal norms.

If m is σ-finite, M(Lp(m), L1(m)) = Lp′

w (m) isometrically (1/p+ 1/p′ = 1).

Therefore, L1Lp(m)(m) = L1(m) ∩ Lp′w (m).

(Fernandez, Mayoral, Naranjo, Saez, S-P, Positivity 2006).

L1Y (η)(m) = L1(m) ∩M(Yb(η), L1(m))

Example.

Lp(m) = f ∈ L0(|λ|) : |f |p ∈ L1(m).

L1Y (η)(m) = L1(m) ∩M(Yb(η), L1(m))

Example.

Lp(m) = f ∈ L0(|λ|) : |f |p ∈ L1(m).

L1Y (η)(m) = L1(m) ∩M(Yb(η), L1(m))

Example.

Lp(m) = f ∈ L0(|λ|) : |f |p ∈ L1(m).

L1Y (η)(m) = L1(m) ∩M(Yb(η), L1(m))

Example.

Lp(m) = f ∈ L0(|λ|) : |f |p ∈ L1(m).

Optimal extensions for operators whichare bounded by a product norm

T : X(µ)→ E is order-w continuous if Tfn → Tf weakly in Ewhenever fn, f ∈ X(µ) with 0 ≤ fn ↑ f µ-a.e.

This property holds for instance if X(µ) is order continuous and T iscontinuous. Note that if T is order-w continuous, then the conditionT (χA) 6= 0 for some A ∈ Σ is equivalent to T being non null.

If T is order-w continuous mT is a vector measure and for everyf ∈ X(µ) we have that f ∈ L1(mT ) with

∫Ω f dmT = Tf .

Let Y (η) be a B.f.s. with mT η. We will say that T isY (η)-extensible if there exists a constant K > 0 such that

‖T (fϕ)‖E ≤ K ‖f‖X(µ)‖ϕ‖Y (η)

for all f ∈ X(µ) and ϕ ∈ S(Σ) ∩ Y (η).

Proposition. The following assertions are equivalent:

(a) T is Y (η)-extensible.

(b) [i] : X(µ)→ L1Y (η)(mT

) is well defined.

(c) P : X(µ)× Yb(η)→ L1(mT

) given by P(f, h) = fh, is well defined.

‖T (fϕ)‖E ≤ K ‖f‖X(µ)‖ϕ‖Y (η)

(b) [i] : X(µ)→ L1Y (η)(mT

) is well defined.

(c) P : X(µ)× Yb(η)→ L1(mT

‖T (fϕ)‖E ≤ K ‖f‖X(µ)‖ϕ‖Y (η)

(b) [i] : X(µ)→ L1Y (η)(mT

) is well defined.

(c) P : X(µ)× Yb(η)→ L1(mT

Theorem

Suppose that T is Y (η)-extensible. If Z(ζ) is a B.f.s. such that ζ µand T factorizes as

X(µ) E-T

HHHj *

Z(ζ)i S

with S being order-w continuous and Y (η)-extensible, then[i] : Z(ζ)→ L1

Y (η)(mT ) is well defined and S(f) = ImT

(f) for all

f ∈ Z(ζ).

Example

Denote by R+ the interval [0,∞), by B the σ-algebra of all Borelsubsets of R+ and by λ the Lebesgue measure on B. Consider theHardy operator S defined on L1 ∩L∞(λ) as S(f)(x) = 1

∫ x0 f(y) dy.

Let ψ : R+ → R+ be an increasing concave map with ψ(0) = 0,ψ(0+) = 0, ψ(∞) =∞. Let

Λψ(λ) = f ∈ L0(λ) : ‖f‖Λψ(λ) =

∫ ∞0

f∗(s) dψ(s) <∞

be the related Lorentz space (f∗ being the decreasing rearrangementof f), which is an order continuous B.f.s. endowed with the norm‖f‖Λψ(λ).

Assume that θψ(t) =∫∞t

ψ′(s)s ds <∞ for all t > 0, where ψ′

denotes the derivative of ψ.

Example

∫ x0 f(y) dy.

Let ψ : R+ → R+ be an increasing concave map with ψ(0) = 0,ψ(0+) = 0, ψ(∞) =∞.

Λψ(λ) = f ∈ L0(λ) : ‖f‖Λψ(λ) =

∫ ∞0

f∗(s) dψ(s) <∞

Example

∫ x0 f(y) dy.

Λψ(λ) = f ∈ L0(λ) : ‖f‖Λψ(λ) =

∫ ∞0

f∗(s) dψ(s) <∞

Example

∫ x0 f(y) dy.

Λψ(λ) = f ∈ L0(λ) : ‖f‖Λψ(λ) =

∫ ∞0

f∗(s) dψ(s) <∞

Example

Consider the Lorentz space Λφ(λ) with φ(t) =∫ t

0 θψ(s) ds for allt > 0.

For f ∈ L1 ∩ L∞(λ) and ϕ ∈ S(B) ∩ Λφ(λ),

‖S(fϕ)‖Λψ(λ) =

∫ ∞0

(S(fϕ)

)∗(s)ψ′(s) ds

≤∫ ∞

ψ′(s)

0f∗(t)ϕ∗(t) dt ds

≤ ‖f‖L∞(λ)

∫ ∞0

ϕ∗(t)

∫ ∞t

ψ′(s)

sds dt ≤ ‖f‖L1∩L∞(λ)‖ϕ‖Λφ(λ),

This shows that S is Λφ(λ)-extensible. Therefore, S factorizesthrough L1

Λφ(λ)(mS ) as in the Theorem and the factorization is

optimal.

Example

∫ ∞0

(S(fϕ)

)∗(s)ψ′(s) ds

≤∫ ∞

ψ′(s)

≤ ‖f‖L∞(λ)

∫ ∞0

ϕ∗(t)

∫ ∞t

ψ′(s)

optimal.

Example

∫ ∞0

(S(fϕ)

)∗(s)ψ′(s) ds

≤∫ ∞

ψ′(s)

≤ ‖f‖L∞(λ)

∫ ∞0

ϕ∗(t)

∫ ∞t

ψ′(s)

optimal.

Example

∫ ∞0

(S(fϕ)

)∗(s)ψ′(s) ds

≤∫ ∞

ψ′(s)

≤ ‖f‖L∞(λ)

∫ ∞0

ϕ∗(t)

∫ ∞t

ψ′(s)

optimal.

Example

∫ ∞0

(S(fϕ)

)∗(s)ψ′(s) ds

≤∫ ∞

ψ′(s)

≤ ‖f‖L∞(λ)

∫ ∞0

ϕ∗(t)

∫ ∞t

ψ′(s)

optimal.

Example

∫ ∞0

(S(fϕ)

)∗(s)ψ′(s) ds

≤∫ ∞

ψ′(s)

≤ ‖f‖L∞(λ)

∫ ∞0

ϕ∗(t)

∫ ∞t

ψ′(s)

optimal.

Examples and applications.

Lp-product extensible operators.

Example

Let (Ω,Σ, µ) be a finite measure space and T : X(µ)→ E a non nullorder-w continuous operator.

Given 1 < p <∞, the operator T is said to be Lp-product extensibleif there exists a constant K > 0 satisfying that

sup‖T (fϕ)‖E : ϕ ∈ S(Σ) ∩BLp′ (µ)

≤ K ‖f‖X(µ)

for all f ∈ X(µ). Calabuig, Galaz, Jimenez, S-P. MathZ 2007.

T is Lp-product extensible if and only if it is Lp′(µ)-extensible.

T can be optimally “extended” preserving the inequality above to thespace L1

Lp′ (µ)(mT ) = L1

p,µ(mT ) by ImT

Pisier’s factorization theorem

Example

Let (Ω,Σ, µ) be a σ-finite measure space, E a Banach space andT : Ls(µ)→ E a non null continuous linear operator, where1 < s <∞. Note that T is order-w continuous (as Ls(µ) is ordercontinuous), Ls(µ) has a weak unit (as µ is a σ-finite measure) andΣLs(µ) = A ∈ Σ : µ(A) <∞.

Then Σ locLs(µ) = Σ. For 1 ≤ p < s, Pisier’s factorization theorem

establishes that T factorizes through a weighted Lorentz spaceLp,1(ωdµ) if and only if it satisfies a lower p-estimate, that is, ifthere exists C > 0 such that( n∑

‖T (fi)‖pE)1/p

≤ C∥∥∥ n∑i=1

|fi|∥∥∥Ls(µ)

for all disjoint f1, ..., fn ∈ Ls(µ).

Example

Let (Ω,Σ, µ) be a σ-finite measure space, E a Banach space andT : Ls(µ)→ E a non null continuous linear operator, where1 < s <∞. Note that T is order-w continuous (as Ls(µ) is ordercontinuous), Ls(µ) has a weak unit (as µ is a σ-finite measure) andΣLs(µ) = A ∈ Σ : µ(A) <∞.

Then Σ locLs(µ) = Σ. For 1 ≤ p < s, Pisier’s factorization theorem

establishes that T factorizes through a weighted Lorentz spaceLp,1(ωdµ) if and only if it satisfies a lower p-estimate, that is, ifthere exists C > 0 such that( n∑

‖T (fi)‖pE)1/p

≤ C∥∥∥ n∑i=1

|fi|∥∥∥Ls(µ)

for all disjoint f1, ..., fn ∈ Ls(µ).

Example

On other hand, this condition is equivalent to the existence of aprobability measure λ on Σ and a constant K > 0 such that

‖T (fg)‖E ≤ K ‖f‖Ls(µ)‖g‖Lt,1(λ)

for all f ∈ Ls(µ) and g a Σ-measurable function with |g| ≤ 1pointwise, where 1

t = 1p −

Then, it follows that T satisfies a lower p-estimate if and only if T isLt,1(λ)-extensible. In this case the Theorem provides the largest“extension” of T preserving the inequality, namely,Im

Lt,1(λ)(mT )→ E.

Example

On other hand, this condition is equivalent to the existence of aprobability measure λ on Σ and a constant K > 0 such that

‖T (fg)‖E ≤ K ‖f‖Ls(µ)‖g‖Lt,1(λ)

for all f ∈ Ls(µ) and g a Σ-measurable function with |g| ≤ 1pointwise, where 1

t = 1p −

Then, it follows that T satisfies a lower p-estimate if and only if T isLt,1(λ)-extensible. In this case the Theorem provides the largest“extension” of T preserving the inequality, namely,Im

Lt,1(λ)(mT )→ E.

p-th power factorable operators.

Example

Let (Ω,Σ) be a measurable space, X(µ) a B.f.s. with a weak unit(so, Σ loc

X(µ) = Σ) and T : X(µ)→ E a non null order-w continuouslinear operator with values in a Banach space E.

Given 0 < r <∞, consider the quasi-B.f.s. (B.f.s. if r ≥ 1)

Xr(µ) = f ∈ L0(µ) : |f |r ∈ X(µ),

equipped with the quasi-norm (norm if r ≥ 1)

‖f‖Xr(µ) = ‖ |f |r‖1/rX(µ).

The factorization of T through Lp(mT ) is related with certainXr(µ)-extensibility.

Proposition. Assume that S(Σ) ∩X(µ) is dense in X(µ) and let1 < p <∞. The following assertions are equivalent:

(a) [i] : X(µ)→ L1(mT

) ∩ Lp(mT

) is well defined.

(b) T is X1

p−1 (µ)-extensible.

Example

Xr(µ) = f ∈ L0(µ) : |f |r ∈ X(µ),

‖f‖Xr(µ) = ‖ |f |r‖1/rX(µ).

(a) [i] : X(µ)→ L1(mT

) ∩ Lp(mT

) is well defined.

(b) T is X1

Example

Xr(µ) = f ∈ L0(µ) : |f |r ∈ X(µ),

‖f‖Xr(µ) = ‖ |f |r‖1/rX(µ).

(a) [i] : X(µ)→ L1(mT

) ∩ Lp(mT

) is well defined.

(b) T is X1

Example

Xr(µ) = f ∈ L0(µ) : |f |r ∈ X(µ),

‖f‖Xr(µ) = ‖ |f |r‖1/rX(µ).

(a) [i] : X(µ)→ L1(mT

) ∩ Lp(mT

) is well defined.

(b) T is X1

Example

Xr(µ) = f ∈ L0(µ) : |f |r ∈ X(µ),

‖f‖Xr(µ) = ‖ |f |r‖1/rX(µ).

(a) [i] : X(µ)→ L1(mT

) ∩ Lp(mT

) is well defined.

(b) T is X1

p-th power factorable operators

Example

Moreover, if (a)-(b) hold, T factorizes as

L1(mT ) ∩ Lp(mT )

X1p−1 (µ)

(mT ),

If E is a 2-concave Banach lattice and T is X(µ)-extensible with mT

being equivalent to µ, then T factorizes through L2(µ) as

X(µ) E-T

HHHj *

L2(µ)Mg S

where Mg is a multiplication operator and S a continuous linear operator.

p-th power factorable operators

Example

Moreover, if (a)-(b) hold, T factorizes as

L1(mT ) ∩ Lp(mT )

X1p−1 (µ)

(mT ),

If E is a 2-concave Banach lattice and T is X(µ)-extensible with mT

being equivalent to µ, then T factorizes through L2(µ) as

X(µ) E-T

HHHj *

L2(µ)Mg S

where Mg is a multiplication operator and S a continuous linear operator.

II Strong factorizations for couples of operators. Delgado, S-P.2011

Strong factorizations for couples of operators.

Definition. Y (µ), Z1(µ), Z2(µ) and X(µ) B.f.s such that Y ⊆ Z1 andZ2 ⊆ X.

Consider T : Y → X and S : Z1 → Z2.

T factorizes strongly through S if there are functions f ∈ Y Z1 andg ∈ ZX2 such that T (y) = gS(fy), i.e the following diagram commutes

Z1(µ)

Y (µ)

Z2(µ),

X(µ)-T

Z(µ), X(µ) B.f.s. Product space ZπX ′

π(h) := inf∑‖fi‖‖gi‖ : |h| ≤

∑|fi||gi|, fi ∈ Z, gi ∈ X ′

Z1(µ)

Y (µ)

Z2(µ),

X(µ)-T

π(h) := inf∑‖fi‖‖gi‖ : |h| ≤

∑|fi||gi|, fi ∈ Z, gi ∈ X ′

Z1(µ)

Y (µ)

Z2(µ),

X(µ)-T

π(h) := inf∑‖fi‖‖gi‖ : |h| ≤

∑|fi||gi|, fi ∈ Z, gi ∈ X ′

Z1(µ)

Y (µ)

Z2(µ),

X(µ)-T

π(h) := inf∑‖fi‖‖gi‖ : |h| ≤

∑|fi||gi|, fi ∈ Z, gi ∈ X ′

Z1(µ)

Y (µ)

Z2(µ),

X(µ)-T

π(h) := inf∑‖fi‖‖gi‖ : |h| ≤

∑|fi||gi|, fi ∈ Z, gi ∈ X ′

Theorem

(Factorization Th.) Let X(µ) an order continuous Banach functionspace with the Fatou property. Let be Z2(µ) and Y (µ) ordercontinuous Banach function spaces, and let Z1(µ) be other Banachfunction space such that Y ⊆ Z1 and Z2 ⊆ X.

Let S : Z1 → Z2 and T : Y → X be operators. Assume that Z2πX′ is

order continuous. The following assertions are equivalent.

(i) (1) There is a function h ∈ Y Z1 such that for all y1, ..., yn ∈ Y (µ) andx′1, ..., x

′2 ∈ X ′,

n∑i=1

〈T (yi), x′i〉 ≤

∥∥∥∥∥n∑i=1

|S(hyi)x′i|

∥∥∥∥∥Z2πX′

(ii) (2) There are two functions f ∈ Y Z1 and g ∈ ZX2 such thatT (y) = gS(fy), i.e. T factorizes strongly through S.

Theorem

′2 ∈ X ′,

n∑i=1

〈T (yi), x′i〉 ≤

∥∥∥∥∥n∑i=1

|S(hyi)x′i|

∥∥∥∥∥Z2πX′

Theorem

′2 ∈ X ′,

n∑i=1

〈T (yi), x′i〉 ≤

∥∥∥∥∥n∑i=1

|S(hyi)x′i|

∥∥∥∥∥Z2πX′

Theorem

′2 ∈ X ′,

n∑i=1

〈T (yi), x′i〉 ≤

∥∥∥∥∥n∑i=1

|S(hyi)x′i|

∥∥∥∥∥Z2πX′

Theorem

′2 ∈ X ′,

n∑i=1

〈T (yi), x′i〉 ≤

∥∥∥∥∥n∑i=1

|S(hyi)x′i|

∥∥∥∥∥Z2πX′

Characteristic inequalities for kernel operators

Corollary. Let Y , X, Z1 and Z2 be Banach function spaces with therequirements of the factorization theorem. Let T : Y → X be anoperator and K a kernel such that SK : Z1 → Z2. The followingassertions are equivalent.

(i) There is a function h ∈ Y Z1 such that for every y1, ..., yn ∈ Y andx′1, ..., x

′n ∈ X ′,

n∑i=1

〈T (yi), x′i〉 ≤

∥∥∥∥∥n∑i=1

|x′i(w)| ·∣∣ ∫

K(w, s)h(s)yi(s)dµ(s)∣∣∥∥∥∥∥Z2πX′

(ii) There is a function f ∈ Y Z1 and a function g ∈ ZX2 such that

T (y)(w) = g(w)

K(w, s)f(s)y(s)dµ(s)

for almost all w ∈ Ω and y ∈ Y .

′n ∈ X ′,

n∑i=1

〈T (yi), x′i〉 ≤

∥∥∥∥∥n∑i=1

|x′i(w)| ·∣∣ ∫

T (y)(w) = g(w)

′n ∈ X ′,

n∑i=1

〈T (yi), x′i〉 ≤

∥∥∥∥∥n∑i=1

|x′i(w)| ·∣∣ ∫

T (y)(w) = g(w)

Corollary. Consider an order continuous Banach function space X(dt)with the Fatou property over the Lebesgue space ([0, 1],Σ, dt). Thefollowing assertions for an operator T : X(dt)→ X(dt) are equivalent.

(1) There is a function h ∈ L∞ such that for every x ∈ X and x′ ∈ X ′,∫T (x)x′dt ≤

∫|x′(w)|

∣∣∣ ∫ w

hx(t)dt∣∣∣dw.

(2) There is a function h ∈ L∞ such that for every simple function x ∈ X,T (x)(w) ≤ |

∫ w0hx(t)dt|, for almost all w ∈ [0, 1].

(3) There are functions f, g ∈ L∞ such that T (x)(w) = g(w)(∫ w

0fxdµ) for

almost all w ∈ [0, 1] and all x ∈ X.

∫|x′(w)|

∣∣∣ ∫ w

hx(t)dt∣∣∣dw.

0fxdµ) for

∫|x′(w)|

∣∣∣ ∫ w

hx(t)dt∣∣∣dw.

0fxdµ) for

∫|x′(w)|

∣∣∣ ∫ w

hx(t)dt∣∣∣dw.

0fxdµ) for

factorization of operators and vector measures

e vector measure

vector measure n

measure space

e banach

y mf y

vector measures n

y semivariation

subspace l1 y

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