En esta charla mostramos que la extensio ́n de un operador definido en un espacio de funciones de Banach es equivalente, bajo ciertos requisitos m ́ınimos, a su factorizacio ́natrave ́sdeunsubespaciodefuncionesintegrablesrespectodela medidavectorialmT asociadaaloperadorT.Adema ́s,estesubespacioparticular determina aquella propiedad del operador que se pretende preservar, en el sentido de queconstituyeeldominioo ́ptimo-elmayorsubespacioalquesepuedeextender-,de manera que la extensio ́n conserva esta propiedad. Esto permite un planteamiento abstracto de ciertos aspectos de la teor ́ıa de dominio o ́ptimo, que se ha desarrollado de manera importante en los u ́ltimos tiempos, principalmente en la direccio ́n de caracterizar cua ́l es el mayor espacio de funciones de Banach al que se pueden extender algunos operadores de importancia en ana ́lisis. Adema ́s, permite clasificar laspropiedadesdelosoperadoresqueseconservanenlaextensio ́nenfuncio ́n delsubespaciodeL1(mT)queconstituyesudominioo ́ptimo.
We compute a low-rank surrogate (response surface) approximation to the solution of stochastic PDE. This is a Karhunen-Loeve/polynomial chaos approximation. After that, to compute required statistics, we sample this cheap surrogate, avoiding very expensive solution of the deterministic problem.
To make Reinforcement Learning Algorithms work in the real-world, one has to get around (what Sutton calls) the "deadly triad": the combination of bootstrapping, function approximation and off-policy evaluation. The first step here is to understand Value Function Vector Space/Geometry and then make one's way into Gradient TD Algorithms (a big breakthrough to overcome the "deadly triad").
We compute a low-rank surrogate (response surface) approximation to the solution of stochastic PDE. This is a Karhunen-Loeve/polynomial chaos approximation. After that, to compute required statistics, we sample this cheap surrogate, avoiding very expensive solution of the deterministic problem.
To make Reinforcement Learning Algorithms work in the real-world, one has to get around (what Sutton calls) the "deadly triad": the combination of bootstrapping, function approximation and off-policy evaluation. The first step here is to understand Value Function Vector Space/Geometry and then make one's way into Gradient TD Algorithms (a big breakthrough to overcome the "deadly triad").
Compactness and homogeneous maps on Banach function spacesesasancpe
The aim of this talk is to show how compact operators between Banach function spaces can be approximated by means of homogeneous maps. After explaining the special characterizations of compact and weakly compact sets that are known for Banach function spaces, we develop a factorization method for describing lattice and topological properties of homogeneous maps by approximating with “simple” homogeneous maps. No approximation properties for the spaces involved are needed. A canonical homogeneous map that is defined as φp(f ) := |f |1/p · ∥f ∥1/p′ from a Banach function space X into its p-th power X[p] plays a meaningful role.
This work has its roots in some classical descriptions of weakly compact subsets of Banach spaces (Grothendieck, Fremlin,...), but particular Banach lattice tools (p-convexification, Maurey-Rosenthal type theorems) are also required.
Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at Uni...Don Sheehy
The Vietoris-Rips filtration is a versatile tool in topological data analysis.
Unfortunately, it is often too large to construct in full.
We show how to construct an $O(n)$-size filtered simplicial complex on an $n$-point metric space such that the persistence diagram is a good approximation to that of the Vietoris-Rips filtration.
The filtration can be constructed in $O(n\log n)$ time.
The constants depend only on the doubling dimension of the metric space and the desired tightness of the approximation.
For the first time, this makes it computationally tractable to approximate the persistence diagram of the Vietoris-Rips filtration across all scales for large data sets.
Our approach uses a hierarchical net-tree to sparsify the filtration.
We can either sparsify the data by throwing out points at larger scales to give a zigzag filtration,
or sparsify the underlying graph by throwing out edges at larger scales to give a standard filtration.
Both methods yield the same guarantees.
Noisy Optimization Convergence Rates (GECCO2013)Jialin LIU
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The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
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RuleML 2015 Constraint Handling Rules - What Else?RuleML
Constraint Handling Rules (CHR) is both a versatile theoretical formalism based on logic and an efficient practical high-level programming language based on rules and constraints.
Procedural knowledge is often expressed by if-then rules, events and actions are related by reaction rules, change is expressed by update rules. Algorithms are often specified using inference rules, rewrite rules, transition rules, sequents, proof rules, or logical axioms. All these kinds of rules can be directly written in CHR. The clean logical semantics of CHR facilitates non-trivial program analysis and transformation. About a dozen implementations of CHR exist in Prolog, Haskell, Java, Javascript and C. Some of them allow to apply millions of rules per second. CHR is also available as WebCHR for online experimentation with more than 40 example programs. More than 200 academic and industrial projects worldwide use CHR, and about 2000 research papers reference it.
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Compactness and homogeneous maps on Banach function spacesesasancpe
The aim of this talk is to show how compact operators between Banach function spaces can be approximated by means of homogeneous maps. After explaining the special characterizations of compact and weakly compact sets that are known for Banach function spaces, we develop a factorization method for describing lattice and topological properties of homogeneous maps by approximating with “simple” homogeneous maps. No approximation properties for the spaces involved are needed. A canonical homogeneous map that is defined as φp(f ) := |f |1/p · ∥f ∥1/p′ from a Banach function space X into its p-th power X[p] plays a meaningful role.
This work has its roots in some classical descriptions of weakly compact subsets of Banach spaces (Grothendieck, Fremlin,...), but particular Banach lattice tools (p-convexification, Maurey-Rosenthal type theorems) are also required.
Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at Uni...Don Sheehy
The Vietoris-Rips filtration is a versatile tool in topological data analysis.
Unfortunately, it is often too large to construct in full.
We show how to construct an $O(n)$-size filtered simplicial complex on an $n$-point metric space such that the persistence diagram is a good approximation to that of the Vietoris-Rips filtration.
The filtration can be constructed in $O(n\log n)$ time.
The constants depend only on the doubling dimension of the metric space and the desired tightness of the approximation.
For the first time, this makes it computationally tractable to approximate the persistence diagram of the Vietoris-Rips filtration across all scales for large data sets.
Our approach uses a hierarchical net-tree to sparsify the filtration.
We can either sparsify the data by throwing out points at larger scales to give a zigzag filtration,
or sparsify the underlying graph by throwing out edges at larger scales to give a standard filtration.
Both methods yield the same guarantees.
Noisy Optimization Convergence Rates (GECCO2013)Jialin LIU
"Noisy optimization convergence rates". Sandra Astete-Morales, Jialin Liu and Olivier Teytaud. (Accepted as short paper) Genetic and Evolutionary Computation Conference (GECCO), 2013.
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
A new practical algorithm for volume estimation using annealing of convex bodiesVissarion Fisikopoulos
We study the problem of estimating the volume of convex polytopes, focusing on H- and V-polytopes, as well as zonotopes. Although a lot of effort is devoted to practical algorithms for H-polytopes there is no such method for the latter two representations. We propose a new, practical algorithm for all representations, which is faster than existing methods. It relies on Hit-and-Run sampling, and combines a new simulated annealing method with the Multiphase Monte Carlo (MMC) approach. Our method introduces the following key features to make it adaptive: (a) It defines a sequence of convex bodies in MMC by introducing a new annealing schedule, whose length is shorter than in previous methods with high probability, and the need of computing an enclosing and an inscribed ball is removed; (b) It exploits statistical properties in rejection-sampling and proposes a better empirical convergence criterion for specifying each step; (c) For zonotopes, it may use a sequence of convex bodies for MMC different than balls, where the chosen body adapts to the input. We offer an open-source, optimized C++ implementation, and analyze its performance to show that it outperforms state-of-the-art software for H-polytopes by Cousins-Vempala (2016) and Emiris-Fisikopoulos (2018), while it undertakes volume computations that were intractable until now, as it is the first polynomial-time, practical method for V-polytopes and zonotopes that scales to high dimensions (currently 100). We further focus on zonotopes, and characterize them by their order (number of generators over dimension), because this largely determines sampling complexity. We analyze a related application, where we evaluate methods of zonotope approximation in engineering.
RuleML 2015 Constraint Handling Rules - What Else?RuleML
Constraint Handling Rules (CHR) is both a versatile theoretical formalism based on logic and an efficient practical high-level programming language based on rules and constraints.
Procedural knowledge is often expressed by if-then rules, events and actions are related by reaction rules, change is expressed by update rules. Algorithms are often specified using inference rules, rewrite rules, transition rules, sequents, proof rules, or logical axioms. All these kinds of rules can be directly written in CHR. The clean logical semantics of CHR facilitates non-trivial program analysis and transformation. About a dozen implementations of CHR exist in Prolog, Haskell, Java, Javascript and C. Some of them allow to apply millions of rules per second. CHR is also available as WebCHR for online experimentation with more than 40 example programs. More than 200 academic and industrial projects worldwide use CHR, and about 2000 research papers reference it.
Нефинансовые меры поддержки внешнеэкономической деятельности и роль ТПП РФ в поддержке предпринимателей.Межправительственные комиссии по торгово-экономическому и научно- техническому сотрудничеству между РФ и иностранными государствами.
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ANTE INTERNATIONAL is one of the fastest growing distribution companies in the Philippines. We offer premier organic health and wellness products and services aimed at helping improve quality of life.
Founded in 2008, SANTE INTERNATIONAL is governed by a formidable team of leaders backed by a combined marketing experience of 50 years. We built and continue to drive business growth through the dedication of our independent distributors.
Driven by our promise to bring best-value wellness products and services in every home, SANTE INTERNATIONAL is committed to equip and empower our distributors who serve as the channel towards a healthy and active lifestyle for our consumers.
Vector measures and classical disjointification methods...esasancpe
In this talk we show how the classical disjointification methods (Bessaga-Pelczynski, Kadecs-Pelczynski) can be applied in the setting of the spaces of p-integrable functions with respect to vector measures. These spaces provide in fact a representation of p-convex order continuous Banach lattices with weak unit; the additional tool of the vector valued integral for each function has already shown to be fruitful for the analysis of these spaces. Consequently, our results can be directly extended to a broad class of Banach lattices.
Vector measures and classical disjointification methodsesasancpe
Vector measures and classical disjointification methods.
In this talk we show how the classical disjointification methods (Bessaga-Pelczynski, Kadecs-Pelczynski) can be applied in the setting of the spaces of p-integrable functions with respect to vector measures. These spaces provide in fact a representation of p-convex order continuous Banach lattices with weak unit; the additional tool of the vector valued integral for each function has already shown to be fruitful for the analysis of these spaces. Consequently, our results can be directly extended to a broad class of Banach lattices.
This presentation is a part of Computer Oriented Numerical Method . Newton-Cotes formulas are an extremely useful and straightforward family of numerical integration techniques.
Operator’s Differential geometry with Riemannian Manifoldsinventionjournals
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In this paper we discussed about the pseudo integral for a measurable function based on a strict pseudo addition and pseudo multiplication. Further more we got several important properties of the pseudo integral of a measurable function based on a strict pseudo addition decomposable measure.
In this paper we discussed about the pseudo integral for a measurable function based on a strict pseudo addition and pseudo multiplication. Further more we got several important properties of the pseudo integral of a measurable function based on a strict pseudo addition decomposable measure.
Computing the volume of a convex body is a fundamental problem in computational geometry and optimization. In this talk we discuss the computational complexity of this problem from a theoretical as well as practical point of view. We show examples of how volume computation appear in applications ranging from combinatorics to algebraic geometry.
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Interestingly, our software provides a framework for exploring theoretical advances since it is believed, and our experiments provide evidence for this belief, that the current asymptotic bounds are unrealistically high.
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Subespacios de funciones integrables respecto de una medidavectorialyextensio ́ndeoperadores
1. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Subespacios de funciones integrables respecto de una
medida vectorial y extensi´on de operadores
Enrique A. S´anchez P´erez
J. M. Calabuig O. Delgado
Instituto de Matem´atica Pura y Aplicada (I.M.P.A.).
Departamento de Matem´atica Aplicada.
Universidad Polit´ecnica de Valencia.
Camino de Vera S/N, 46022, Valencia.
X EARCO
Mallorca, Mayo 2007
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
2. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
En esta charla mostramos que la extensi´on de un operador definido en un espacio de
funciones de Banach es equivalente, bajo ciertos requisitos m´ınimos, a su
factorizaci´on a trav´es de un subespacio de funciones integrables respecto de la
medida vectorial mT asociada al operador T. Adem´as, este subespacio particular
determina aquella propiedad del operador que se pretende preservar, en el sentido de
que constituye el dominio ´optimo -el mayor subespacio al que se puede extender-, de
manera que la extensi´on conserva esta propiedad. Esto permite un planteamiento
abstracto de ciertos aspectos de la teor´ıa de dominio ´optimo, que se ha desarrollado
de manera importante en los ´ultimos tiempos, principalmente en la direcci´on de
caracterizar cu´al es el mayor espacio de funciones de Banach al que se pueden
extender algunos operadores de importancia en an´alisis. Adem´as, permite clasificar
las propiedades de los operadores que se conservan en la extensi´on en funci´on
del subespacio de L1(mT ) que constituye su dominio ´optimo.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
3. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Index of the talk
1 Notation and preliminaries
2 Vector measures and spaces of integrable functions
3 A factorization theorem
4 An application: extensions of operators on p-convex B. f. s.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
4. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Index of the talk
1 Notation and preliminaries
2 Vector measures and spaces of integrable functions
3 A factorization theorem
4 An application: extensions of operators on p-convex B. f. s.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
5. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Index of the talk
1 Notation and preliminaries
2 Vector measures and spaces of integrable functions
3 A factorization theorem
4 An application: extensions of operators on p-convex B. f. s.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
6. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Index of the talk
1 Notation and preliminaries
2 Vector measures and spaces of integrable functions
3 A factorization theorem
4 An application: extensions of operators on p-convex B. f. s.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
7. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
FIRST PART
Notation and preliminaries
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
8. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
General notation
(Ω,Σ,µ) will be a finite measure space,
E be a Banach space and E its topological dual,
BE the unit ball of E,
P(A) will represent the set of partitions π of A into finite number of disjoint
measurable sets,
m : Σ → E will be a (countably additive) vector measure,
The semivariation of m over A ∈ Σ is defined by
m (A) = sup
x ∈BE
| m,x |(A) = sup
x ∈BE
sup
π∈P(A)
∑
B∈π
| m,x (B)|,
with the usual notation
m,x (B) = m(B),x for each B ∈ Σ,
A ∈ Σ is called m-null if m (A) = 0.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
9. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
General notation
(Ω,Σ,µ) will be a finite measure space,
E be a Banach space and E its topological dual,
BE the unit ball of E,
P(A) will represent the set of partitions π of A into finite number of disjoint
measurable sets,
m : Σ → E will be a (countably additive) vector measure,
The semivariation of m over A ∈ Σ is defined by
m (A) = sup
x ∈BE
| m,x |(A) = sup
x ∈BE
sup
π∈P(A)
∑
B∈π
| m,x (B)|,
with the usual notation
m,x (B) = m(B),x for each B ∈ Σ,
A ∈ Σ is called m-null if m (A) = 0.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
10. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Remark
In this talk we assume that µ and m are mutually absolutely continuous.
In particular, µ can be a Rybakov measure; recall that a Rybakov measure for a vector
measure m is a scalar measure ν defined as the variation of a measure m,x , where
x ∈ E , whenever m is absolutely continuous with respect to ν.
A Rybakov measure always exists for every vector measure m (see [5, IX.2.2]).
Banach function (sub)spaces.
Let L0(µ) be the space of all measurable real functions on Ω.
1 We will call Banach function space to any Banach space X(µ) ⊆ L0(µ) with norm
· X(µ) satisfying that
1) if f ∈ L0(µ), g ∈ X(µ) with |f| ≤ |g| µ-a.e. then f ∈ X(µ) and f X(µ) ≤ g X(µ).
2) For every A ∈ Σ, the characteristic function χA belongs to X(µ).
2 By a Banach function subspace of X(µ) we mean a Banach function space
continuously contained in X(µ), with the same order structure (allowing different
norms).
3 A Banach function space is order continuous if order bounded increasing
sequences are convergent in norm.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
11. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Remark
In this talk we assume that µ and m are mutually absolutely continuous.
In particular, µ can be a Rybakov measure; recall that a Rybakov measure for a vector
measure m is a scalar measure ν defined as the variation of a measure m,x , where
x ∈ E , whenever m is absolutely continuous with respect to ν.
A Rybakov measure always exists for every vector measure m (see [5, IX.2.2]).
Banach function (sub)spaces.
Let L0(µ) be the space of all measurable real functions on Ω.
1 We will call Banach function space to any Banach space X(µ) ⊆ L0(µ) with norm
· X(µ) satisfying that
1) if f ∈ L0(µ), g ∈ X(µ) with |f| ≤ |g| µ-a.e. then f ∈ X(µ) and f X(µ) ≤ g X(µ).
2) For every A ∈ Σ, the characteristic function χA belongs to X(µ).
2 By a Banach function subspace of X(µ) we mean a Banach function space
continuously contained in X(µ), with the same order structure (allowing different
norms).
3 A Banach function space is order continuous if order bounded increasing
sequences are convergent in norm.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
12. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
L1(m)L1(m)L1(m) spaces.
A function f : Ω → R is said to be integrable with respect to the measure m if
1 for each x ∈ E we have that f ∈ L1( m,x ),
2 for each A ∈ Σ there exists xA ∈ E such that
xA,x =
A
fd m,x for every x ∈ E .
The element xA is usually denoted by A f dm. The space of the classes (equality
m-almost everywhere) of these functions is denoted by L1(m). The expression
f m = sup
x ∈BE
|f|d| m,x | for each f ∈ L1
(m),
defines a lattice norm on L1(m) for which L1(m) is an order continuous Banach
function space.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
13. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
The indefinite integral mf : Σ → E of a function f ∈ L1(m) is defined by
mf (A) =
A
fdm, A ∈ Σ.
The Orlicz-Pettis Theorem ensures that mf is again a countably additive vector
measure. An equivalent norm for L1(m) is given by
|||f|||m = sup
A∈Σ A
fdm
E
for each f ∈ L1
(m).
Moreover |||f|||L1(m) ≤ f L1(m) ≤ 2|||f|||L1(m) ([5, Ch. I.1.11]).
Lp(m)Lp(m)Lp(m) spaces.
Let 1 ≤ p < ∞. A measurable function f : Ω → R is said to be p-integrable if |f|p is
integrable with respect to m. The expression.
f Lp(m) = sup
x ∈BE
|f|p
d| m,x |
1
p
for each f ∈ Lp
(m),
defines a lattice norm on Lp(m) ([6]). The space Lp(m) is an order continuous Banach
function subspace of L1(m).
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
14. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
The indefinite integral mf : Σ → E of a function f ∈ L1(m) is defined by
mf (A) =
A
fdm, A ∈ Σ.
The Orlicz-Pettis Theorem ensures that mf is again a countably additive vector
measure. An equivalent norm for L1(m) is given by
|||f|||m = sup
A∈Σ A
fdm
E
for each f ∈ L1
(m).
Moreover |||f|||L1(m) ≤ f L1(m) ≤ 2|||f|||L1(m) ([5, Ch. I.1.11]).
Lp(m)Lp(m)Lp(m) spaces.
Let 1 ≤ p < ∞. A measurable function f : Ω → R is said to be p-integrable if |f|p is
integrable with respect to m. The expression.
f Lp(m) = sup
x ∈BE
|f|p
d| m,x |
1
p
for each f ∈ Lp
(m),
defines a lattice norm on Lp(m) ([6]). The space Lp(m) is an order continuous Banach
function subspace of L1(m).
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
15. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
The indefinite integral mf : Σ → E of a function f ∈ L1(m) is defined by
mf (A) =
A
fdm, A ∈ Σ.
The Orlicz-Pettis Theorem ensures that mf is again a countably additive vector
measure. An equivalent norm for L1(m) is given by
|||f|||m = sup
A∈Σ A
fdm
E
for each f ∈ L1
(m).
Moreover |||f|||L1(m) ≤ f L1(m) ≤ 2|||f|||L1(m) ([5, Ch. I.1.11]).
Lp(m)Lp(m)Lp(m) spaces.
Let 1 ≤ p < ∞. A measurable function f : Ω → R is said to be p-integrable if |f|p is
integrable with respect to m. The expression.
f Lp(m) = sup
x ∈BE
|f|p
d| m,x |
1
p
for each f ∈ Lp
(m),
defines a lattice norm on Lp(m) ([6]). The space Lp(m) is an order continuous Banach
function subspace of L1(m).
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
16. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
SECOND PART
Vector measures and spaces of integrable functions.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
17. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Y(µ)Y(µ)Y(µ) semivariation.
Let m : Σ → E be a (countably additive) vector measure and Y(µ) a Banach function
space. The Y(µ)-semivariation m Y(µ) of m is defined by
m Y(µ) := sup
n
∑
i=1
αAi
m(Ai )
E
:
n
∑
i=1
αAi
χAi
∈ BY(µ) .
L1
Y(µ)(m)L1
Y(µ)(m)L1
Y(µ)(m) spaces.
We define L1
Y(µ)(m) as the space of all (classes of m-a.e. equal) functions f of L1(m)
such that the vector measure associated to m, mf , has finite Y(µ)-semivariation
endowed with the norm
f L1
Y(µ)
(m) := mf Y(µ) = sup
ϕ simple
ϕ∈BY(µ)
fϕdm
E
.
L1
Y(µ)(m) is always a Banach function subspace of L1(m).
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
18. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Y(µ)Y(µ)Y(µ) semivariation.
Let m : Σ → E be a (countably additive) vector measure and Y(µ) a Banach function
space. The Y(µ)-semivariation m Y(µ) of m is defined by
m Y(µ) := sup
n
∑
i=1
αAi
m(Ai )
E
:
n
∑
i=1
αAi
χAi
∈ BY(µ) .
L1
Y(µ)(m)L1
Y(µ)(m)L1
Y(µ)(m) spaces.
We define L1
Y(µ)(m) as the space of all (classes of m-a.e. equal) functions f of L1(m)
such that the vector measure associated to m, mf , has finite Y(µ)-semivariation
endowed with the norm
f L1
Y(µ)
(m) := mf Y(µ) = sup
ϕ simple
ϕ∈BY(µ)
fϕdm
E
.
L1
Y(µ)(m) is always a Banach function subspace of L1(m).
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
19. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Example
Some particular subspaces of L1(m) that are nowadays well known can be
represented in terms of the spaces L1
Y(µ)(m).
1 The semivariation of a vector measure can be computed by
m = sup
n
∑
i=1
αAi
m(Ai )
E
:
n
∑
i=1
αAi
χAi
∈ BL∞(µ) = m L∞(µ),
where µ is a finite positive measure equivalent to m (see [5, Ch.I]). Thus, for any
function f ∈ L1(m), mf = mf L∞(µ) = f L1
L∞(µ)
(m).
Hence, L1(m) = L1
L∞(µ)(m).
2 Let 1 < p < ∞ and 1/p +1/p = 1. It is known that for any function f ∈ Lp(m), the
expression
mf Lp (m)
= sup
ϕ simple
ϕ∈B
Lp (m)
fϕdm
E
equals the norm f L
p
w (m)
of a function f belonging to L
p
w (m) ( [7] ).
In fact, it can be proved that L
p
w (m) = L1
Lp (m)
(m).
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
20. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Example
Some particular subspaces of L1(m) that are nowadays well known can be
represented in terms of the spaces L1
Y(µ)(m).
1 The semivariation of a vector measure can be computed by
m = sup
n
∑
i=1
αAi
m(Ai )
E
:
n
∑
i=1
αAi
χAi
∈ BL∞(µ) = m L∞(µ),
where µ is a finite positive measure equivalent to m (see [5, Ch.I]). Thus, for any
function f ∈ L1(m), mf = mf L∞(µ) = f L1
L∞(µ)
(m).
Hence, L1(m) = L1
L∞(µ)(m).
2 Let 1 < p < ∞ and 1/p +1/p = 1. It is known that for any function f ∈ Lp(m), the
expression
mf Lp (m)
= sup
ϕ simple
ϕ∈B
Lp (m)
fϕdm
E
equals the norm f L
p
w (m)
of a function f belonging to L
p
w (m) ( [7] ).
In fact, it can be proved that L
p
w (m) = L1
Lp (m)
(m).
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
21. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Example
Some particular subspaces of L1(m) that are nowadays well known can be
represented in terms of the spaces L1
Y(µ)(m).
1 The semivariation of a vector measure can be computed by
m = sup
n
∑
i=1
αAi
m(Ai )
E
:
n
∑
i=1
αAi
χAi
∈ BL∞(µ) = m L∞(µ),
where µ is a finite positive measure equivalent to m (see [5, Ch.I]). Thus, for any
function f ∈ L1(m), mf = mf L∞(µ) = f L1
L∞(µ)
(m).
Hence, L1(m) = L1
L∞(µ)(m).
2 Let 1 < p < ∞ and 1/p +1/p = 1. It is known that for any function f ∈ Lp(m), the
expression
mf Lp (m)
= sup
ϕ simple
ϕ∈B
Lp (m)
fϕdm
E
equals the norm f L
p
w (m)
of a function f belonging to L
p
w (m) ( [7] ).
In fact, it can be proved that L
p
w (m) = L1
Lp (m)
(m).
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
22. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Example
Some particular subspaces of L1(m) that are nowadays well known can be
represented in terms of the spaces L1
Y(µ)(m).
1 The semivariation of a vector measure can be computed by
m = sup
n
∑
i=1
αAi
m(Ai )
E
:
n
∑
i=1
αAi
χAi
∈ BL∞(µ) = m L∞(µ),
where µ is a finite positive measure equivalent to m (see [5, Ch.I]). Thus, for any
function f ∈ L1(m), mf = mf L∞(µ) = f L1
L∞(µ)
(m).
Hence, L1(m) = L1
L∞(µ)(m).
2 Let 1 < p < ∞ and 1/p +1/p = 1. It is known that for any function f ∈ Lp(m), the
expression
mf Lp (m)
= sup
ϕ simple
ϕ∈B
Lp (m)
fϕdm
E
equals the norm f L
p
w (m)
of a function f belonging to L
p
w (m) ( [7] ).
In fact, it can be proved that L
p
w (m) = L1
Lp (m)
(m).
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
23. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
THIRD PART
A factorization theorem.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
24. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Let X(µ) be an order continuous Banach function space and let T : X(µ) → E be an
operator. As usual, we define the (countably additive) vector measure
mT : Σ → E by mT (A) := T(χA).
An operator T is µ-determined if the set of mT -null sets equals the set of µ-null sets.
Let Y(µ) be a Banach function space over µ containing the set of the simple functions.
Y(µ)Y(µ)Y(µ)-extensible operators.
A µ-determined operator T : X(µ) → E is said to be Y(µ)-extensible if there is a
constant K > 0 such that
T(fϕ) ≤ K f X(µ) ϕ Y(µ)
for every function f ∈ X(µ) and every simple function ϕ.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
25. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Let X(µ) be an order continuous Banach function space and let T : X(µ) → E be an
operator. As usual, we define the (countably additive) vector measure
mT : Σ → E by mT (A) := T(χA).
An operator T is µ-determined if the set of mT -null sets equals the set of µ-null sets.
Let Y(µ) be a Banach function space over µ containing the set of the simple functions.
Y(µ)Y(µ)Y(µ)-extensible operators.
A µ-determined operator T : X(µ) → E is said to be Y(µ)-extensible if there is a
constant K > 0 such that
T(fϕ) ≤ K f X(µ) ϕ Y(µ)
for every function f ∈ X(µ) and every simple function ϕ.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
26. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Let X(µ) be an order continuous Banach function space and let T : X(µ) → E be an
operator. As usual, we define the (countably additive) vector measure
mT : Σ → E by mT (A) := T(χA).
An operator T is µ-determined if the set of mT -null sets equals the set of µ-null sets.
Let Y(µ) be a Banach function space over µ containing the set of the simple functions.
Y(µ)Y(µ)Y(µ)-extensible operators.
A µ-determined operator T : X(µ) → E is said to be Y(µ)-extensible if there is a
constant K > 0 such that
T(fϕ) ≤ K f X(µ) ϕ Y(µ)
for every function f ∈ X(µ) and every simple function ϕ.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
27. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Theorem (First part)
Let X(µ) be an order continuous Banach function space with and let Y(µ) be a
Banach function space verifying that the simple functions are dense. Given a
µ-determined operator T : X(µ) → E the following assertions are equivalent
1 T is Y(µ)-extensible.
2 The bounded linear map T can be factored through the space L1
Y(µ)(mT ) as
X(µ)
T
//
i
$$
E
L1
Y(µ)(mT )
IY(µ)
;;
where the integration map IY(µ) is also Y(µ)-extensible.
3 There is a constant K > 0 verifying that for every f ∈ L1(m)
(mT )f Y(µ) ≤ K f X(µ)
where (mT )f : Σ → E is the vector measure given by (mT )f (A) = A fdmT .
4 X(µ)·Y(µ) ⊆ L1(mT ).
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
28. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Theorem (First part)
Let X(µ) be an order continuous Banach function space with and let Y(µ) be a
Banach function space verifying that the simple functions are dense. Given a
µ-determined operator T : X(µ) → E the following assertions are equivalent
1 T is Y(µ)-extensible.
2 The bounded linear map T can be factored through the space L1
Y(µ)(mT ) as
X(µ)
T
//
i
$$
E
L1
Y(µ)(mT )
IY(µ)
;;
where the integration map IY(µ) is also Y(µ)-extensible.
3 There is a constant K > 0 verifying that for every f ∈ L1(m)
(mT )f Y(µ) ≤ K f X(µ)
where (mT )f : Σ → E is the vector measure given by (mT )f (A) = A fdmT .
4 X(µ)·Y(µ) ⊆ L1(mT ).
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
29. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Theorem (Second part, optimal domain)
Moreover, if Z(µ) is any other order continuous Banach function space such that
X(µ) ⊆ Z(µ) with continuous inclusion,
T can be extended to Z(µ) and this extension is Y(µ)-extensible,
then Z(µ) ⊆ L1
Y(µ)(mT ) with continuous inclusion.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
30. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Example
Assume that T is continuous. In this case taking into account that for each f ∈ X(µ)
and each ϕ simple function
T(fϕ) E ≤ K fϕ X(µ) ≤ K f X(µ) ϕ L∞(µ),
we obtain that T is L∞(µ)-extensible. Reciprocally if T is L∞(µ)-extensible then taking
ϕ = χΩ we obtain that T is continuous. Applying Theorem 3 we obtain that the optimal
domain is L1
L∞(µ)(m) = L1(m) so this result includes the case studied by Curbera G.P.
and Ricker W. in [2].
Example ([1])
In [1] the authors introduce the notion of Lp-product extensible operator and define the
spaces L1
p,µ (m). In this article they prove that the bigger function space (i.e. the optimal
domain) to which an Lp-product extensible operator T can be extended preserving this
property is exactly the space L1
p,µ (mT ). Actually the notion of Lp-product extensible
operator coincides with our definition of Y(µ)-extensible operator with Y(µ) = Lp(µ) for
1 < p < ∞. In this case the space L1
p,µ (mT ) is now L1
Lp(µ)(mT ). Hence Theorem 6 in [1]
is a particular case of Theorem 3.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
31. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Example
Assume that T is continuous. In this case taking into account that for each f ∈ X(µ)
and each ϕ simple function
T(fϕ) E ≤ K fϕ X(µ) ≤ K f X(µ) ϕ L∞(µ),
we obtain that T is L∞(µ)-extensible. Reciprocally if T is L∞(µ)-extensible then taking
ϕ = χΩ we obtain that T is continuous. Applying Theorem 3 we obtain that the optimal
domain is L1
L∞(µ)(m) = L1(m) so this result includes the case studied by Curbera G.P.
and Ricker W. in [2].
Example ([1])
In [1] the authors introduce the notion of Lp-product extensible operator and define the
spaces L1
p,µ (m). In this article they prove that the bigger function space (i.e. the optimal
domain) to which an Lp-product extensible operator T can be extended preserving this
property is exactly the space L1
p,µ (mT ). Actually the notion of Lp-product extensible
operator coincides with our definition of Y(µ)-extensible operator with Y(µ) = Lp(µ) for
1 < p < ∞. In this case the space L1
p,µ (mT ) is now L1
Lp(µ)(mT ). Hence Theorem 6 in [1]
is a particular case of Theorem 3.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
32. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
FOURTH PART
An application: extension of operators on p-convex B. f. s.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
33. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
A Banach lattice E is p-convex if there is a constant K such that for each finite set
x1,...,xn ∈ E,
(
n
∑
i=1
|xi |p
)1/p
≤ K(
n
∑
i=1
xi
p
)1/p
.
Let T : E → F be an operator, where F is a lattice. T is p-concave if there is a
constant K such that for each finite family x1,...,xn ∈ E,
(
n
∑
i=1
T(xi ) p
)1/p
≤ K (
n
∑
i=1
|xi |p
)1/p
.
If X(µ) a Banach function space we define its p-th power X(µ)[p] as the space
X(µ)[p] := {|f|p
: f ∈ X(µ)}.
If X(µ) is p-convex, X(µ)[p] is a Banach function space over µ with the quasi norm
g X(µ)[p]
:= |g|1/p p
X(µ)
, g ∈ X(µ)[p],
that in this case is equivalent to a norm.
If µ is a finite measure, then the inclusion X(µ) ⊆ X(µ)[p] is well-defined and
continuous.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
34. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
A Banach lattice E is p-convex if there is a constant K such that for each finite set
x1,...,xn ∈ E,
(
n
∑
i=1
|xi |p
)1/p
≤ K(
n
∑
i=1
xi
p
)1/p
.
Let T : E → F be an operator, where F is a lattice. T is p-concave if there is a
constant K such that for each finite family x1,...,xn ∈ E,
(
n
∑
i=1
T(xi ) p
)1/p
≤ K (
n
∑
i=1
|xi |p
)1/p
.
If X(µ) a Banach function space we define its p-th power X(µ)[p] as the space
X(µ)[p] := {|f|p
: f ∈ X(µ)}.
If X(µ) is p-convex, X(µ)[p] is a Banach function space over µ with the quasi norm
g X(µ)[p]
:= |g|1/p p
X(µ)
, g ∈ X(µ)[p],
that in this case is equivalent to a norm.
If µ is a finite measure, then the inclusion X(µ) ⊆ X(µ)[p] is well-defined and
continuous.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
35. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
A Banach lattice E is p-convex if there is a constant K such that for each finite set
x1,...,xn ∈ E,
(
n
∑
i=1
|xi |p
)1/p
≤ K(
n
∑
i=1
xi
p
)1/p
.
Let T : E → F be an operator, where F is a lattice. T is p-concave if there is a
constant K such that for each finite family x1,...,xn ∈ E,
(
n
∑
i=1
T(xi ) p
)1/p
≤ K (
n
∑
i=1
|xi |p
)1/p
.
If X(µ) a Banach function space we define its p-th power X(µ)[p] as the space
X(µ)[p] := {|f|p
: f ∈ X(µ)}.
If X(µ) is p-convex, X(µ)[p] is a Banach function space over µ with the quasi norm
g X(µ)[p]
:= |g|1/p p
X(µ)
, g ∈ X(µ)[p],
that in this case is equivalent to a norm.
If µ is a finite measure, then the inclusion X(µ) ⊆ X(µ)[p] is well-defined and
continuous.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
36. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
A Banach lattice E is p-convex if there is a constant K such that for each finite set
x1,...,xn ∈ E,
(
n
∑
i=1
|xi |p
)1/p
≤ K(
n
∑
i=1
xi
p
)1/p
.
Let T : E → F be an operator, where F is a lattice. T is p-concave if there is a
constant K such that for each finite family x1,...,xn ∈ E,
(
n
∑
i=1
T(xi ) p
)1/p
≤ K (
n
∑
i=1
|xi |p
)1/p
.
If X(µ) a Banach function space we define its p-th power X(µ)[p] as the space
X(µ)[p] := {|f|p
: f ∈ X(µ)}.
If X(µ) is p-convex, X(µ)[p] is a Banach function space over µ with the quasi norm
g X(µ)[p]
:= |g|1/p p
X(µ)
, g ∈ X(µ)[p],
that in this case is equivalent to a norm.
If µ is a finite measure, then the inclusion X(µ) ⊆ X(µ)[p] is well-defined and
continuous.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
37. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Let X(µ) be an order continuous p-convex Banach function space and E a
p-concave Banach lattice. Consider a positive operator T : X(µ) → E. Then it is
known that there is a function h belonging to the dual of X(µ)[p] such that
T(f) ≤ K |f|p
hdµ
1/p
, f ∈ X(µ).
We can assume also that h > 0 µ-a.e.
We can use this inequality to generate a norm function by
q∗
(ϕ) := sup{ T(fϕ) : |f|p
hdµ ≤ 1}, ϕ simple.
By using a density argument we can generate a function space G(µ) satisfying
that
fϕ dmT ≤ Kq∗
(ϕ) |f|p
hdµ
1/p
.
Now we can apply the factorization theorem to this inequality. We obtain in this
way an extension of T to the subspace L1
G(µ)(mT ) that is optimal in the sense that
L1
G(µ)(mT ) is the biggest space to which T can be extended still satisfying the
inequality.
The subspace L1
G(µ)(mT ) is p-convex.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores
38. Notation and preliminaries
Vector measures and spaces of integrable functions
A factorization theorem
An application: extensions of operators on p-convex B. f. s.
Some references
Calabuig J.M., Galaz F., Jim´enez Fern´andez E. and S´anchez P´erez E.A., Strong factorization
of operators on spaces of vector measure integrable functions and unconditional convergence
of series. To appear in Mathematische Zeitschrift.
Curbera, G.P. and Ricker, W.J.: Optimal domains for the kernel operator associated with
Sobolev’s inequality , Studia Math., 158(2), (2003), 131-152 (see also Corrigenda in the same
journal , 170, (2005), 217-218).
Delgado, O.: Optimal domains for kernel operators on [0,∞)×[0,∞), Studia Math. 174 (2006),
131-145.
Delgado, O. and Soria, J.: Optimal domain for the Hardy operator, J. Funct. Anal. 244 (2007),
119-133.
Diestel J. and Uhl J.J., Vector Measures, Math. Surveys, vol. 15, Amer. Math. Soc.,
Providence, RI, 1977.
Fern´andez, A., Mayoral, F., Naranjo, F., S´aez, C. and S´anchez-P´erez, E.A.: Spaces of
p-integrable functions with respect to a vector measure., Positivity, 10(2006), 1-16.
S´anchez P´erez E.A., Compactness arguments for spaces of p-integrable functions with
respect to a vector measure and factorization of operators through Lebesgue-Bochner
spaces. Illinois J. Math. 45,3(2001), 907-923.
Enrique A. S´anchez P´erez Subespacios y extensiones de operadores