FACTORIZATION OF OPERATORS AND VECTOR MEASURES

FACTORIZATION OF OPERATORS
AND VECTOR MEASURES
Enrique A. Sánchez Pérez
Instituto de Matemática Pura y Aplicada
Universidad Politécnica de Valencia

Factorization of operators and vector measures
An operator on a Banach function space -and also on more general
classes of Banach spaces of functions- always deﬁnes a vector mea-
sure. A vector measure is always related in a natural way with a
Banach function space -its space of integrable functions-. This is
the starting point of a methodological purpose that deals with vector
measures, operators and spaces of integrable functions from a uni-
ﬁed point of view. It provides information about the structure of the
operators and is often presented in terms of factorization theorems.
There are many old and new results that can be considered as part
of 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 factorization
of operators, the analysis of the topological properties of the integra-
tion map -compactness, weak compactness, complete continuity-, and
its geometric properties -q-convexity and q-concavity- related to the
structure of the space L1(m).

Riesz Representation Theorem
Radon-Nikod´ym Theorem

VECTOR VALUED VERSIONS:
Riesz Representation Theorem ⇒ Bartle, Dunford, Schwartz
T : C(K) → E, T(f) = fd mT .
Radon-Nikod´ym Theorem

VECTOR VALUED VERSIONS:
Riesz Representation Theorem ⇒ Bartle, Dunford, Schwartz
T : C(K) → E, T(f) = fd mT .
Radon-Nikod´ym Theorem ⇒ Diestel, Uhl
T : L1
(µ) → E, T(f) = φfd µ.

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)

L1
of a vector measure
Extensions and factorization of operators. mT (A) := T(χA)

L1
of a vector measure
Extensions and factorization of operators. mT (A) := T(χA)
Optimal domain theorem
X(µ) EET
rrrj ¨¨¨B
L1
(mT )
i ImT

Radon-Nikod´ym derivatives for vector measures
n m ⇔ n(A) = A h dm (Musial)
Where is h? (Calabuig, Gregori, S-P)

Radon-Nikod´ym derivatives for vector measures
n m ⇔ n(A) = A h dm (Musial)
Where is h? (Calabuig, Gregori, S-P)
L1
(m) EEIm
r
rrj ¨
¨¨B
L1
(n)
i In

Factorizing operators through spaces of multiplication
operators (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 a
δ-ring deﬁned by means of semivariations.

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

2.- Optimal extensions for operators which are bounded by a product
norm.
3.- Examples and Applications.

Subspaces of L1
(m) and generalized semivariations

Subspaces of L1
(Ω, Σ, η) measure space. E Banach. Y (η) B.f.s. n : Σ → E vector
measure, η n.

Subspaces of L1
measure, η n.
Y (η)-semivariation of n
n Y (η) = sup
Ω
ϕ dn
E
: ϕ ∈ S(Σ) ∩ BY (η) .

Subspaces of L1
measure, η n.
n Y (η) = sup
Ω
ϕ dn
E
: ϕ ∈ S(Σ) ∩ BY (η) .
Subspace L1
Y (η)(m) = f ∈ L1(m) : mf Y (η) < ∞ .
Norm f L1
Y (η)
(m) = max { f m, mf Y (η)}.

Subspaces of L1
measure, η n.
n Y (η) = sup
Ω
ϕ dn
E
: ϕ ∈ S(Σ) ∩ BY (η) .
Subspace L1
Y (η)(m) = f ∈ L1(m) : mf Y (η) < ∞ .
Norm f L1
Y (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).

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

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

Subspaces of L1
measure, η n.
fg L1(n).
It can be written as the dual of a product space (Delgado, S-P, IEOT
2010).

Subspaces of L1
measure, η n.
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).

Proposition. The equality
L1
Y (η)(m) = L1
(m) ∩ M(Yb(η), L1
(m))
holds and mf Y (η) = f M(Yb(η),L1(m)) for all f ∈ L1
Y (η)(m).

L1
Y (η)(m) = L1
(m) ∩ M(Yb(η), L1
(m))
Y (η)(m).
Example.
Y (η) = Lp
(η) with 1 ≤ p < ∞.
L1
Lp(η)(m) = L1
(m) ∩ M(Lp
(η), L1
(m)) isometrically.

L1
Y (η)(m) = L1
(m) ∩ M(Yb(η), L1
(m))
Y (η)(m).
Example.
Y (η) = Lp
(η) with 1 ≤ p < ∞.
L1
Lp(η)(m) = L1
(m) ∩ M(Lp
(η), L1
(m)) isometrically.
If η is ﬁnite L1
Lp(η)(m) = M(Lp
(η), L1
(m)). It coincides with L1
p ,η(m)
(1
p + 1
p = 1), given by all functions f ∈ L1
(m) such that p -semivariation
of mf with respect to η is ﬁnite.

L1
Y (η)(m) = L1
(m) ∩ M(Yb(η), L1
(m))
Y (η)(m).
Example.
Y (η) = Lp
(η) with 1 ≤ p < ∞.
L1
Lp(η)(m) = L1
(m) ∩ M(Lp
(η), L1
(m)) isometrically.
If η is finite L1
Lp(η)(m) = M(Lp
(η), L1
(m)). It coincides with L1
p ,η(m)
(1
p + 1
p = 1), given by all functions f ∈ L1
(m) such that p -semivariation
of mf with respect to η is finite. That is,
mf p ,η = sup
π∈P(Ω)
sup
x∗∈BE∗
A∈π
| A
fdx∗
m|p
η(A)p −1
1/p
< ∞,
(Calabuig, Galaz, Jiménez, S-P, MathZ 2007).

L1
Y (η)(m) = L1
(m) ∩ M(Yb(η), L1
(m))
Y (η)(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)}.

L1
Y (η)(m) = L1
(m) ∩ M(Yb(η), L1
(m))
Y (η)(m).
Example.
1 ≤ p < ∞, Lp
(m), i.e.
Lp
(m) = {f ∈ L0
(|λ|) : |f|p
∈ L1
(m)}.
L1
Lp(m)(m) = L1
(m) ∩ M(Lp
(m), L1
(m)) with equal norms.

L1
Y (η)(m) = L1
(m) ∩ M(Yb(η), L1
(m))
Y (η)(m).
Example.
1 ≤ p < ∞, Lp
(m), i.e.
Lp
(m) = {f ∈ L0
(|λ|) : |f|p
∈ L1
(m)}.
L1
Lp(m)(m) = L1
(m) ∩ M(Lp
(m), L1
If m is σ-ﬁnite, M(Lp
(m), L1
(m)) = Lp
w (m) isometrically (1/p + 1/p = 1).

L1
Y (η)(m) = L1
(m) ∩ M(Yb(η), L1
(m))
Y (η)(m).
Example.
1 ≤ p < ∞, Lp
(m), i.e.
Lp
(m) = {f ∈ L0
(|λ|) : |f|p
∈ L1
(m)}.
L1
Lp(m)(m) = L1
(m) ∩ M(Lp
(m), L1
(m), L1
(m)) = Lp
Therefore, L1
Lp(m)(m) = L1
(m) ∩ Lp
w (m).

L1
Y (η)(m) = L1
(m) ∩ M(Yb(η), L1
(m))
Y (η)(m).
Example.
1 ≤ p < ∞, Lp
(m), i.e.
Lp
(m) = {f ∈ L0
(|λ|) : |f|p
∈ L1
(m)}.
L1
Lp(m)(m) = L1
(m) ∩ M(Lp
(m), L1
(m), L1
(m)) = Lp
Therefore, L1
Lp(m)(m) = L1
(m) ∩ Lp
w (m).
(Fern´andez, Mayoral, Naranjo, S´aez, S-P, Positivity 2006).

Optimal extensions for operators which
are bounded by a product norm

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

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

This property holds for instance if X(µ) is order continuous and T is
continuous. Note that if T is order-w continuous, then the condition
T(χA) = 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 every
f ∈ 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 is
Y (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).

Let Y (η) be a B.f.s. with mT η. We will say that T is
Y (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) → L1
Y (η)(mT
) is well deﬁned.
(c) P : X(µ) × Yb(η) → L1
(mT
) given by P(f, h) = fh, is well deﬁned.

Theorem
Suppose that T is Y (η)-extensible. If Z(ζ) is a B.f.s. such that ζ µ
and T factorizes as
X(µ) EET
rrrj ¨¨¨B
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 Borel
subsets of R+ and by λ the Lebesgue measure on B. Consider the
Hardy operator S deﬁned on L1 ∩ L∞(λ) as S(f)(x) = 1
x
x
0 f(y) dy.

Example
x
x
0 f(y) dy.
Let ψ: R+ → R+ be an increasing concave map with ψ(0) = 0,
ψ(0+) = 0, ψ(∞) = ∞.

Example
x
x
0 f(y) dy.
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 rearrangement
of f), which is an order continuous B.f.s. endowed with the norm
f Λψ(λ).

Example
x
x
0 f(y) dy.
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 rearrangement
of 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
Consider the Lorentz space Λφ(λ) with φ(t) =
t
0 θψ(s) ds for all
t > 0.

Example
t
t > 0.
For f ∈ L1 ∩ L∞(λ) and ϕ ∈ S(B) ∩ Λφ(λ),

Example
t
t > 0.
S(fϕ) Λψ(λ) =
∞
0
S(fϕ)
∗
(s)ψ (s) ds

Example
t
t > 0.
S(fϕ) Λψ(λ) =
∞
0
S(fϕ)
∗
(s)ψ (s) ds
≤
∞
0
ψ (s)
s
s
0
f∗
(t)ϕ∗
(t) dt ds

Example
t
t > 0.
S(fϕ) Λψ(λ) =
∞
0
S(fϕ)
∗
(s)ψ (s) ds
≤
∞
0
ψ (s)
s
s
0
f∗
(t)ϕ∗
(t) dt ds
≤ f L∞(λ)
∞
0
ϕ∗
(t)
∞
t
ψ (s)
s
ds dt ≤ f L1∩L∞(λ) ϕ Λφ(λ),

Example
t
t > 0.
S(fϕ) Λψ(λ) =
∞
0
S(fϕ)
∗
(s)ψ (s) ds
≤
∞
0
ψ (s)
s
s
0
f∗
(t)ϕ∗
(t) dt ds
≤ f L∞(λ)
∞
0
ϕ∗
(t)
∞
t
ψ (s)
s
ds dt ≤ f L1∩L∞(λ) ϕ Λφ(λ),
This shows that S is Λφ(λ)-extensible. Therefore, S factorizes
through L1
Λφ(λ)(mS ) as in the Theorem and the factorization is
optimal.

Lp
-product extensible operators.
Example
Let (Ω, Σ, µ) be a ﬁnite measure space and T : X(µ) → E a non null
order-w continuous operator.
Given 1 < p < ∞, the operator T is said to be Lp-product extensible
if 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, Jim´enez, 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 the
space 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 and
T : Ls(µ) → E a non null continuous linear operator, where
1 < s < ∞. Note that T is order-w continuous (as Ls(µ) is order
continuous), Ls(µ) has a weak unit (as µ is a σ-ﬁnite measure) and
ΣLs(µ) = {A ∈ Σ : µ(A) < ∞}.

Example
Let (Ω, Σ, µ) be a σ-finite measure space, E a Banach space and
T : Ls(µ) → E a non null continuous linear operator, where
1 < s < ∞. Note that T is order-w continuous (as Ls(µ) is order
continuous), Ls(µ) has a weak unit (as µ is a σ-finite measure) and
ΣLs(µ) = {A ∈ Σ : µ(A) < ∞}.
Then Σloc
Ls(µ) = Σ. For 1 ≤ p < s, Pisier’s factorization theorem
establishes that T factorizes through a weighted Lorentz space
Lp,1(ωdµ) if and only if it satisfies a lower p-estimate, that is, if
there exists C > 0 such that
n
i=1
T(fi) p
E
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 a
probability 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| ≤ 1
pointwise, where 1
t = 1
p − 1
s .

Example
On other hand, this condition is equivalent to the existence of a
probability 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| ≤ 1
pointwise, where 1
t = 1
p − 1
s .
Then, it follows that T satisﬁes a lower p-estimate if and only if T is
Lt,1(λ)-extensible. In this case the Theorem provides the largest
“extension” of T preserving the inequality, namely,
ImT
: L1
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, Σloc
X(µ) = Σ) and T : X(µ) → E a non null order-w continuous
linear operator with values in a Banach space E.

Example
(so, Σloc
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/r
X(µ).

Example
(so, Σloc
Xr
(µ) = {f ∈ L0
(µ) : |f|r
∈ X(µ)},
f Xr(µ) = |f|r 1/r
X(µ).
The factorization of T through Lp(mT ) is related with certain
Xr(µ)-extensibility.

Example
(so, Σloc
Xr
(µ) = {f ∈ L0
(µ) : |f|r
∈ X(µ)},
f Xr(µ) = |f|r 1/r
X(µ).
Proposition. Assume that S(Σ) ∩ X(µ) is dense in X(µ) and let
1 < p < ∞. The following assertions are equivalent:
(a) [i]: X(µ) → L1
(mT
) ∩ Lp
(mT
) is well deﬁned.

Example
(so, Σloc
Xr
(µ) = {f ∈ L0
(µ) : |f|r
∈ X(µ)},
f Xr(µ) = |f|r 1/r
X(µ).
Proposition. Assume that S(Σ) ∩ X(µ) is dense in X(µ) and let
1 < p < ∞. The following assertions are equivalent:
(a) [i]: X(µ) → L1
(mT
) ∩ Lp
(mT
) is well deﬁned.
(b) T is X
1
p−1 (µ)-extensible.

p-th power factorable operators
Example
Moreover, if (a)-(b) hold, T factorizes as
L1(mT ) ∩ Lp(mT )
X(µ)
i
c
ImT
E
i
L1
X
1
p−1 (µ)
(mT ),
EE
T
T

p-th power factorable operators
Example
Moreover, if (a)-(b) hold, T factorizes as
L1(mT ) ∩ Lp(mT )
X(µ)
i
c
ImT
E
i
L1
X
1
p−1 (µ)
(mT ),
EE
T
T
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) EET
rrrj ¨¨¨B
L2
(µ)
Mg S
where Mg is a multiplication operator and S a continuous linear operator.

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 and
Z2 ⊆ X.

Z2 ⊆ X.
Consider T : Y → X and S : Z1 → Z2.

Z2 ⊆ X.
T factorizes strongly through S if there are functions f ∈ Y Z1 and
g ∈ ZX
2 such that T(y) = gS(fy), i.e the following diagram commutes

Z2 ⊆ X.
g ∈ ZX
Z1(µ)
Y (µ)
f
c
g
E
S
Z2(µ),
X(µ)E
T
T

Z2 ⊆ X.
g ∈ ZX
Z1(µ)
Y (µ)
f
c
g
E
S
Z2(µ),
X(µ)E
T
T
Z(µ), X(µ) B.f.s. Product space ZπX
π(h) := inf{ fi gi : |h| ≤ |fi||gi|, fi ∈ Z, gi ∈ X }

Theorem

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

Theorem
Let S : Z1 → Z2 and T : Y → X be operators. Assume that Z2πX is
order continuous. The following assertions are equivalent.

Theorem
(i) (1) There is a function h ∈ Y Z1
such that for all y1, ..., yn ∈ Y (µ) and
x1, ..., x2 ∈ X ,
n
i=1
T(yi), xi ≤
n
i=1
|S(hyi)xi|
Z2πX
.

Theorem
(i) (1) There is a function h ∈ Y Z1
such that for all y1, ..., yn ∈ Y (µ) and
x1, ..., x2 ∈ X ,
n
i=1
T(yi), xi ≤
n
i=1
|S(hyi)xi|
Z2πX
.
(ii) (2) There are two functions f ∈ Y Z1
and g ∈ ZX
2 such that
T(y) = gS(fy), i.e. T factorizes strongly through S.

Characteristic inequalities for kernel operators
Corollary. Let Y , X, Z1 and Z2 be Banach function spaces with the
requirements of the factorization theorem. Let T : Y → X be an
operator and K a kernel such that SK : Z1 → Z2. The following
assertions are equivalent.

(i) There is a function h ∈ Y Z1
such that for every y1, ..., yn ∈ Y and
x1, ..., xn ∈ X ,
n
i=1
T(yi), xi ≤
n
i=1
|xi(w)| ·
Ω
K(w, s)h(s)yi(s)dµ(s)
Z2πX
.

(i) There is a function h ∈ Y Z1
such that for every y1, ..., yn ∈ Y and
x1, ..., xn ∈ X ,
n
i=1
T(yi), xi ≤
n
i=1
|xi(w)| ·
Ω
K(w, s)h(s)yi(s)dµ(s)
Z2πX
.
(ii) There is a function f ∈ Y Z1
and a function g ∈ ZX
2 such that
T(y)(w) = g(w)
Ω
K(w, s)f(s)y(s)dµ(s)
for almost all w ∈ Ω and y ∈ Y .

Corollary. Consider an order continuous Banach function space X(dt)
with the Fatou property over the Lebesgue space ([0, 1], Σ, dt). The
following 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
0
hx(t)dt dw.

T(x)x dt ≤ |x (w)|
w
0
hx(t)dt dw.
such that for every simple function x ∈ X,
T(x)(w) ≤ |
w
0
hx(t)dt|, for almost all w ∈ [0, 1].

T(x)x dt ≤ |x (w)|
w
0
hx(t)dt dw.
such that for every simple function x ∈ X,
T(x)(w) ≤ |
w
0
hx(t)dt|, for almost all w ∈ [0, 1].
(3) There are functions f, g ∈ L∞
such that T(x)(w) = g(w)(
w
0
fxdµ) for
almost all w ∈ [0, 1] and all x ∈ X.

FACTORIZATION OF OPERATORS AND VECTOR MEASURES

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