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.
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.
Convex Analysis and Duality (based on "Functional Analysis and Optimization" ...Katsuya Ito
In this presentation, we explain the monograph ”Functional Analysis and Optimization” by Kazufumi Ito
https://kito.wordpress.ncsu.edu/files/2018/04/funa3.pdf
Our goal in this presentation is to
-Understand the basic notions of functional analysis
lower-semicontinuous, subdifferential, conjugate functional
- Understand the formulation of duality problem
primal (P), perturbed (Py), and dual (P∗) problem
-Understand the primal-dual relationships
inf(P)≤sup(P∗), inf(P) = sup(P∗), inf supL≤sup inf L
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.
Convex Analysis and Duality (based on "Functional Analysis and Optimization" ...Katsuya Ito
In this presentation, we explain the monograph ”Functional Analysis and Optimization” by Kazufumi Ito
https://kito.wordpress.ncsu.edu/files/2018/04/funa3.pdf
Our goal in this presentation is to
-Understand the basic notions of functional analysis
lower-semicontinuous, subdifferential, conjugate functional
- Understand the formulation of duality problem
primal (P), perturbed (Py), and dual (P∗) problem
-Understand the primal-dual relationships
inf(P)≤sup(P∗), inf(P) = sup(P∗), inf supL≤sup inf L
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.
Subespacios de funciones integrables respecto de una medidavectorialye...esasancpe
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.
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.
Subespacios de funciones integrables respecto de una medidavectorialye...esasancpe
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.
Operator’s Differential geometry with Riemannian Manifoldsinventionjournals
In this paper some fundamental theorems , operators differential geometry – with operator Riemannian geometry to pervious of differentiable manifolds which are used in an essential way in basic concepts of Spectrum of Discrete , bounded Riemannian geometry, we study the defections, examples of the problem of differentially projection mapping parameterization system on dimensional manifolds .
An Analysis and Study of Iteration Proceduresijtsrd
In computational mathematics, an iterative method is a scientific technique that utilizes an underlying speculation to produce a grouping of improving rough answers for a class of issues, where the n th estimate is gotten from the past ones. A particular execution of an iterative method, including the end criteria, is a calculation of the iterative method. An iterative method is called joined if the relating grouping meets for given starting approximations. A scientifically thorough combination investigation of an iterative method is typically performed notwithstanding, heuristic based iterative methods are additionally normal. This Research provides a survey of iteration procedures that have been used to obtain fixed points for maps satisfying a variety of contractive conditions. Dr. R. B. Singh | Shivani Tomar ""An Analysis and Study of Iteration Procedures"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-4 , June 2019, URL: https://www.ijtsrd.com/papers/ijtsrd23715.pdf
Paper URL: https://www.ijtsrd.com/mathemetics/computational-science/23715/an-analysis-and-study-of-iteration-procedures/dr-r-b-singh
A polynomial interpolation algorithm is developed using the Newton's divided-difference interpolating polynomials. The definition of monotony of a function is then used to define the least degree of the polynomial to make efficient and consistent the interpolation in the discrete given function. The relation between the order of monotony of a particular function and the degree of the interpolating polynomial is justified, analyzing the relation between the derivatives of such function and the truncation error expression. In this algorithm there is not matter about the number and the arrangement of the data points, neither if the points are regularly spaced or not. The algorithm thus defined can be used to make interpolations in functions of one and several dependent variables. The algoritm automatically select the data points nearest to the point where an interpolation is desired, following the criterion of symmetry. Indirectly, the algorithm also select the number of data points, which is a unity higher than the order of the used polynomial, following the criterion of monotony. Finally, the complete algoritm is presented and subroutines in fortran code is exposed as an addendum. Notice that there is not the degree of the interpolating polynomial within the arguments of such subroutines.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
The French Revolution Class 9 Study Material pdf free download
Compactness and homogeneous maps on Banach function spaces
1. Compactness and homogeneous maps on Banach function spaces
Compactness and homogeneous maps
on Banach function spaces
Enrique A. S´anchez P´erez
U.P. Valencia
Based on a joint work with Pilar Rueda
Encuentros Murcia-Valencia 2013
Alcoy
17-18 Octubre 2013
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 1 / 37
2. Compactness and homogeneous maps on Banach function spaces
Abstract
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.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 2 / 37
3. Compactness and homogeneous maps on Banach function spaces
Figure : Edu.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 3 / 37
4. Compactness and homogeneous maps on Banach function spaces
Factorization of compact operators through a universal space:
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 4 / 37
5. Compactness and homogeneous maps on Banach function spaces
Factorization of compact operators through a universal space:
Aliprantis, Burkinshaw, Positive operators.
Th.5.5. An operator between Banach spaces is compact if and only if
it factors with compact factors through a closed subspace of c0.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 4 / 37
6. Compactness and homogeneous maps on Banach function spaces
Factorization of compact operators through a universal space:
Aliprantis, Burkinshaw, Positive operators.
Th.5.5. An operator between Banach spaces is compact if and only if
it factors with compact factors through a closed subspace of c0.
Johnson, Factoring compact operators, Israel J. Math. 9 (1971)
Figiel, Factorization of compact operators and applications to the
approximation property, Studia Math. (1973)
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 4 / 37
7. Compactness and homogeneous maps on Banach function spaces
Factorization of compact operators through a universal space:
Aliprantis, Burkinshaw, Positive operators.
Th.5.5. An operator between Banach spaces is compact if and only if
it factors with compact factors through a closed subspace of c0.
Johnson, Factoring compact operators, Israel J. Math. 9 (1971)
Figiel, Factorization of compact operators and applications to the
approximation property, Studia Math. (1973)
Uniform factorization:
Aron, Lindstr¨om, Ruess, Ryan, Uniform factorization for compact sets
of operators. Proceedings AMS (1999)
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 4 / 37
8. Compactness and homogeneous maps on Banach function spaces
Domination of compact operators:
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 5 / 37
9. Compactness and homogeneous maps on Banach function spaces
Domination of compact operators:
Aliprantis, Burkinshaw, Positive operators. Th.5.4. (Terziogly): An
operator T : E → F is compact if and only if there is a norm-null sequence
(x∗
n )n ⊂ E∗ such that
T(x) ≤ sup
n
| x, x∗
n |, x ∈ E.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 5 / 37
10. Compactness and homogeneous maps on Banach function spaces
Basic definitions
1 Basic definitions
2 Relevant subsets and maps
3 Domination of homogeneous maps
4 Compact linear operators
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 6 / 37
11. Compactness and homogeneous maps on Banach function spaces
Basic definitions
If φ : U → E is an homogeneous map from an homogeneous subset U
of a Banach space F, we will call φ := supx∈U∩BF
φ(x) the norm
of φ on U.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 7 / 37
12. Compactness and homogeneous maps on Banach function spaces
Basic definitions
If φ : U → E is an homogeneous map from an homogeneous subset U
of a Banach space F, we will call φ := supx∈U∩BF
φ(x) the norm
of φ on U.
Let (Ω, Σ, µ) be a finite measure space, and L0(µ) the space of
equivalence classes of µ-measurable functions.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 7 / 37
13. Compactness and homogeneous maps on Banach function spaces
Basic definitions
A real Banach space X(µ) of (equivalence classes of) µ-measurable
functions in L0(µ) is a Banach function space over µ if X(µ) ⊂ L1(µ)
and contains all the characteristic functions of measurable sets, and
the norm · X(µ) satisfies that if f ∈ L0(µ), g ∈ X(µ) and |f | ≤ |g|
µ–a.e. then f ∈ X(µ) and
f X(µ) ≤ g X(µ).
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 8 / 37
14. Compactness and homogeneous maps on Banach function spaces
Basic definitions
A real Banach space X(µ) of (equivalence classes of) µ-measurable
functions in L0(µ) is a Banach function space over µ if X(µ) ⊂ L1(µ)
and contains all the characteristic functions of measurable sets, and
the norm · X(µ) satisfies that if f ∈ L0(µ), g ∈ X(µ) and |f | ≤ |g|
µ–a.e. then f ∈ X(µ) and
f X(µ) ≤ g X(µ).
We will simply write X if the measure µ is already fixed.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 8 / 37
15. Compactness and homogeneous maps on Banach function spaces
Basic definitions
A real Banach space X(µ) of (equivalence classes of) µ-measurable
functions in L0(µ) is a Banach function space over µ if X(µ) ⊂ L1(µ)
and contains all the characteristic functions of measurable sets, and
the norm · X(µ) satisfies that if f ∈ L0(µ), g ∈ X(µ) and |f | ≤ |g|
µ–a.e. then f ∈ X(µ) and
f X(µ) ≤ g X(µ).
We will simply write X if the measure µ is already fixed.
The inclusions L∞(µ) ⊂ X(µ) ⊂ L1(µ) are always continuous, since
they are positive.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 8 / 37
16. Compactness and homogeneous maps on Banach function spaces
Basic definitions
A Banach function space X is order continuous if decreasing sequences of
functions that converges to zero µ-a.e., converges to zero also in the norm.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 9 / 37
17. Compactness and homogeneous maps on Banach function spaces
Basic definitions
A Banach function space X is order continuous if decreasing sequences of
functions that converges to zero µ-a.e., converges to zero also in the norm.
A Banach function space X is Fatou if the pointwise limit of an increasing
sequence of functions whose norms are uniformly bounded belongs to X.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 9 / 37
18. Compactness and homogeneous maps on Banach function spaces
Basic definitions
A Banach function space X is order continuous if decreasing sequences of
functions that converges to zero µ-a.e., converges to zero also in the norm.
A Banach function space X is Fatou if the pointwise limit of an increasing
sequence of functions whose norms are uniformly bounded belongs to X.
If X(µ) and Z(µ) are Banach function spaces and X ⊆ Z, we define the
space of multiplication operators XZ as the space of (classes of)
measurable functions defining operators from X to Z by pointwise
multiplication, endowed with the operator norm.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 9 / 37
19. Compactness and homogeneous maps on Banach function spaces
Basic definitions
A Banach function space X is order continuous if decreasing sequences of
functions that converges to zero µ-a.e., converges to zero also in the norm.
A Banach function space X is Fatou if the pointwise limit of an increasing
sequence of functions whose norms are uniformly bounded belongs to X.
If X(µ) and Z(µ) are Banach function spaces and X ⊆ Z, we define the
space of multiplication operators XZ as the space of (classes of)
measurable functions defining operators from X to Z by pointwise
multiplication, endowed with the operator norm.
A particular case is given by X := XL1
, that is called the K¨othe dual of
X.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 9 / 37
20. Compactness and homogeneous maps on Banach function spaces
Basic definitions
Let 0 < p < ∞. If f is a measurable function, we write f 1/p for the
measurable function sign{f }|f |1/p.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 10 / 37
21. Compactness and homogeneous maps on Banach function spaces
Basic definitions
Let 0 < p < ∞. If f is a measurable function, we write f 1/p for the
measurable function sign{f }|f |1/p.
If X(µ) is a Banach function space, the p-convexification of X is
defined as
X[p] := {f ∈ L0
(µ) : |f |1/p
∈ X}
that is a quasi-Banach function space over µ when endowed with the
seminorm f X[p]
:= |f |1/p p
X , f ∈ X[p].
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 10 / 37
22. Compactness and homogeneous maps on Banach function spaces
Basic definitions
Let 0 < p < ∞. If f is a measurable function, we write f 1/p for the
measurable function sign{f }|f |1/p.
If X(µ) is a Banach function space, the p-convexification of X is
defined as
X[p] := {f ∈ L0
(µ) : |f |1/p
∈ X}
that is a quasi-Banach function space over µ when endowed with the
seminorm f X[p]
:= |f |1/p p
X , f ∈ X[p].
For 1 ≤ p < ∞ the inclusion X[1/p] → X always holds and has
relevant properties.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 10 / 37
23. Compactness and homogeneous maps on Banach function spaces
Basic definitions
Let 0 < p < ∞. If f is a measurable function, we write f 1/p for the
measurable function sign{f }|f |1/p.
If X(µ) is a Banach function space, the p-convexification of X is
defined as
X[p] := {f ∈ L0
(µ) : |f |1/p
∈ X}
that is a quasi-Banach function space over µ when endowed with the
seminorm f X[p]
:= |f |1/p p
X , f ∈ X[p].
For 1 ≤ p < ∞ the inclusion X[1/p] → X always holds and has
relevant properties.
Fact: for each f ∈ X(µ)[1/p], its norm can be computed as
f X[1/p]
:= suph∈BX[1/p ]
fh X .
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 10 / 37
24. Compactness and homogeneous maps on Banach function spaces
Basic definitions
Let f (x) := 3Sin(6x)x2.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 11 / 37
25. Compactness and homogeneous maps on Banach function spaces
Basic definitions
Let f (x) := 3Sin(6x)x2.
Figure : Functions f and f 1/2
.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 11 / 37
26. Compactness and homogeneous maps on Banach function spaces
Basic definitions
Figure : f 1/2
, f 1/3
, f 1/4
and f 1/5
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 12 / 37
27. Compactness and homogeneous maps on Banach function spaces
Relevant subsets and maps
1 Basic definitions
2 Relevant subsets and maps
3 Domination of homogeneous maps
4 Compact linear operators
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 13 / 37
28. Compactness and homogeneous maps on Banach function spaces
Relevant subsets and maps
Approximation of compact homogeneous operators
Idea: Using as elementary items for the approximation homogeneous maps
φ between Banach function spaces that can be factored as φ = φ2 ◦ φ1,
where
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 14 / 37
29. Compactness and homogeneous maps on Banach function spaces
Relevant subsets and maps
Approximation of compact homogeneous operators
Idea: Using as elementary items for the approximation homogeneous maps
φ between Banach function spaces that can be factored as φ = φ2 ◦ φ1,
where
1) φ1 carries norm bounded sets to “almost” order bounded sets, and
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 14 / 37
30. Compactness and homogeneous maps on Banach function spaces
Relevant subsets and maps
Approximation of compact homogeneous operators
Idea: Using as elementary items for the approximation homogeneous maps
φ between Banach function spaces that can be factored as φ = φ2 ◦ φ1,
where
1) φ1 carries norm bounded sets to “almost” order bounded sets, and
2) φ2 carries order bounded sets to compact sets.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 14 / 37
31. Compactness and homogeneous maps on Banach function spaces
Relevant subsets and maps
Approximation of compact homogeneous operators
Idea: Using as elementary items for the approximation homogeneous maps
φ between Banach function spaces that can be factored as φ = φ2 ◦ φ1,
where
1) φ1 carries norm bounded sets to “almost” order bounded sets, and
2) φ2 carries order bounded sets to compact sets.
More concretely...
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 14 / 37
32. Compactness and homogeneous maps on Banach function spaces
Relevant subsets and maps
1) A set B ⊂ Z(ν) is approximately order bounded if for each ε > 0
there is f ∈ Z(ν) such that B ⊆ [−f , f ] + εBZ .
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 15 / 37
33. Compactness and homogeneous maps on Banach function spaces
Relevant subsets and maps
1) A set B ⊂ Z(ν) is approximately order bounded if for each ε > 0
there is f ∈ Z(ν) such that B ⊆ [−f , f ] + εBZ .
A (bounded homogeneous) operator T : X(µ) → Z(ν) is semicompact if
T(BX ) is approximately order bounded.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 15 / 37
34. Compactness and homogeneous maps on Banach function spaces
Relevant subsets and maps
1) A set B ⊂ Z(ν) is approximately order bounded if for each ε > 0
there is f ∈ Z(ν) such that B ⊆ [−f , f ] + εBZ .
A (bounded homogeneous) operator T : X(µ) → Z(ν) is semicompact if
T(BX ) is approximately order bounded.
A set B ⊆ Z(ν) is uniformly ν-absolutely continuous if
lim
ν(A)→0
sup
f ∈B
f χA Z = 0.
These sets are also sometimes called uniformly µ-integrable sets.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 15 / 37
35. Compactness and homogeneous maps on Banach function spaces
Relevant subsets and maps
1) A set B ⊂ Z(ν) is approximately order bounded if for each ε > 0
there is f ∈ Z(ν) such that B ⊆ [−f , f ] + εBZ .
A (bounded homogeneous) operator T : X(µ) → Z(ν) is semicompact if
T(BX ) is approximately order bounded.
A set B ⊆ Z(ν) is uniformly ν-absolutely continuous if
lim
ν(A)→0
sup
f ∈B
f χA Z = 0.
These sets are also sometimes called uniformly µ-integrable sets.
Uniformly ν-absolutely continuous maps: maps T : X(µ) → Z(ν) that
satisfy that T(BX ) is a uniformly ν-absolutely continuous subset of Z.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 15 / 37
36. Compactness and homogeneous maps on Banach function spaces
Relevant subsets and maps
1) A set B ⊂ Z(ν) is approximately order bounded if for each ε > 0
there is f ∈ Z(ν) such that B ⊆ [−f , f ] + εBZ .
A (bounded homogeneous) operator T : X(µ) → Z(ν) is semicompact if
T(BX ) is approximately order bounded.
A set B ⊆ Z(ν) is uniformly ν-absolutely continuous if
lim
ν(A)→0
sup
f ∈B
f χA Z = 0.
These sets are also sometimes called uniformly µ-integrable sets.
Uniformly ν-absolutely continuous maps: maps T : X(µ) → Z(ν) that
satisfy that T(BX ) is a uniformly ν-absolutely continuous subset of Z. For
order continuous Banach function spaces, the class of uniformly
ν-absolutely continuous operators and semicompact operators coincide.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 15 / 37
37. Compactness and homogeneous maps on Banach function spaces
Relevant subsets and maps
2) A continuous linear operator T : X(µ) → E is said to be essentially
compact if the set {T(χA) : A ∈ Σ} is relatively (norm) compact in E.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 16 / 37
38. Compactness and homogeneous maps on Banach function spaces
Relevant subsets and maps
2) A continuous linear operator T : X(µ) → E is said to be essentially
compact if the set {T(χA) : A ∈ Σ} is relatively (norm) compact in E.
Essential compactness of an operator T is equivalent to the fact that the
restriction of T to the space L∞(µ) is compact.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 16 / 37
39. Compactness and homogeneous maps on Banach function spaces
Relevant subsets and maps
2) A continuous linear operator T : X(µ) → E is said to be essentially
compact if the set {T(χA) : A ∈ Σ} is relatively (norm) compact in E.
Essential compactness of an operator T is equivalent to the fact that the
restriction of T to the space L∞(µ) is compact.
For order continuous Banach function spaces this is equivalent to the
operator being AM-compact, i.e. T maps order bounded sets to relatively
compact sets.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 16 / 37
40. Compactness and homogeneous maps on Banach function spaces
Domination of homogeneous maps
1 Basic definitions
2 Relevant subsets and maps
3 Domination of homogeneous maps
4 Compact linear operators
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 17 / 37
41. Compactness and homogeneous maps on Banach function spaces
Domination of homogeneous maps
Theorem
Let E be a Banach space, and let Y (ν) and Z(ν) be Banach function
spaces such that Y (ν) ⊆ Z(ν). Let U ⊆ E be an homogeneous set, and
φ : U → Y (ν) and P : U → Z(ν) be bounded homogeneous maps.
Assume also that Y Z has the Fatou property and (Y Z ) is order
continuous. Then the following statements are equivalent.
(i) There is a constant K > 0 such that for every x1, ..., xn ∈ U and
A1, ..., An ∈ Σ,
n
i=1
|P(xi )|χAi L1(ν) ≤ K
n
i=1
|φ(xi )|χAi (Y Z ) .
(ii) There is a function g ∈ KBY Z such that
|P(x)| ≤ g|φ(x)| µ − a.e.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 18 / 37
42. Compactness and homogeneous maps on Banach function spaces
Domination of homogeneous maps
(iii) There is a function g ∈ KBY Z such that for every x ∈ U there is a
function hx ∈ BL∞(ν) depending only on x/ x such that
P(x) = g · hx · φ(x). In other words, P factorizes through the
homogeneous map ˆφ given by ˆφ(x) := hx · φ(x) as
U
P //
ˆφ !!
Z(ν)
Y (ν)
g
;;
In this case we will say that P is strongly dominated by φ.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 19 / 37
43. Compactness and homogeneous maps on Banach function spaces
Domination of homogeneous maps
Example
Take U = X(µ) = Z and Y = X[1/p] for 1 < p < ∞. Let 0 ≤ h ∈ X[1/p].
The homogeneous map P : X → X given by
P(f ) := sgn(f ) f |f |
f ∧ h , f ∈ X, is an example of a map satisfying
(ii) in the previous theorem, since it factors as P = ip ◦ φ, for φ = P,
where ip : X[1/p] → X is the inclusion map.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 20 / 37
44. Compactness and homogeneous maps on Banach function spaces
Domination of homogeneous maps
Theorem
Let X(µ) be an order continuous Banach function space with the Fatou
property and 1 < p < ∞. Let P : X(µ) → X(µ) and
φ : X(µ) → X(µ)[1/p] be bounded homogeneous maps. If P is strongly
dominated by φ then P(BX ) is a uniformly µ-absolutely continuous set.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 21 / 37
45. Compactness and homogeneous maps on Banach function spaces
Domination of homogeneous maps
If B ⊂ X is an homogeneous subset, we define the characteristic
homogeneous operator φB : X → X by φB(f ) = f if f ∈ B and φB(f ) = 0
otherwise.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 22 / 37
46. Compactness and homogeneous maps on Banach function spaces
Domination of homogeneous maps
If B ⊂ X is an homogeneous subset, we define the characteristic
homogeneous operator φB : X → X by φB(f ) = f if f ∈ B and φB(f ) = 0
otherwise.
If 1 ≤ p < ∞, we define the homogeneous map φp : X(µ) → X[1/p] by
φp(·) := (·)1/p (·) 1/p .
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 22 / 37
47. Compactness and homogeneous maps on Banach function spaces
Domination of homogeneous maps
If B ⊂ X is an homogeneous subset, we define the characteristic
homogeneous operator φB : X → X by φB(f ) = f if f ∈ B and φB(f ) = 0
otherwise.
If 1 ≤ p < ∞, we define the homogeneous map φp : X(µ) → X[1/p] by
φp(·) := (·)1/p (·) 1/p .
Theorem
Let X(µ) be an order continuous space with the Fatou property. Let
1 < p < ∞. An homogeneous set B ⊂ X(µ) is order bounded if and only
if its characteristic homogeneous operator φB is strongly dominated by φp.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 22 / 37
48. Compactness and homogeneous maps on Banach function spaces
Domination of homogeneous maps
Theorem
Let X(µ) be an order continuous Banach function space and let E be a
Banach space. The following statements for a bounded homogeneous
operator S : X(µ) → E and a function 0 < g ∈ X[1/p ] are equivalent.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 23 / 37
49. Compactness and homogeneous maps on Banach function spaces
Domination of homogeneous maps
Theorem
Let X(µ) be an order continuous Banach function space and let E be a
Banach space. The following statements for a bounded homogeneous
operator S : X(µ) → E and a function 0 < g ∈ X[1/p ] are equivalent.
(i) For all functions h ∈ X[1/p], S(hp) ≤ hp 1/p
X · gh X .
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 23 / 37
50. Compactness and homogeneous maps on Banach function spaces
Domination of homogeneous maps
Theorem
Let X(µ) be an order continuous Banach function space and let E be a
Banach space. The following statements for a bounded homogeneous
operator S : X(µ) → E and a function 0 < g ∈ X[1/p ] are equivalent.
(i) For all functions h ∈ X[1/p], S(hp) ≤ hp 1/p
X · gh X .
(ii) The homogeneous operator S factorizes through
X(µ)
S //
φp
E
X(µ)[1/p]
g
// X(µ) ,
R
OO
where R is a bounded homogeneous operator with R ≤ 1.
In this case, S satisfies that limµ(A)→0 supf ∈BX
S(f χA) = 0.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 23 / 37
51. Compactness and homogeneous maps on Banach function spaces
Domination of homogeneous maps
Lemma
Let E be a Banach space. The norm limit T of a sequence (Sn)n of
homogeneous bounded operators Sn : X(µ) → E satisfying
lim
µ(A)→0
sup
f ∈BX
Sn(f χA) = 0
for each n, is bounded, homogeneous and satisfies
lim
µ(A)→0
sup
f ∈BX
T(f χA) = 0.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 24 / 37
52. Compactness and homogeneous maps on Banach function spaces
Domination of homogeneous maps
Theorem
∗ Let X(µ) be a Fatou order continuous Banach function space and let
T : X(µ) → E be a bounded homogeneous operator. Each of the
followings statements implies the next one.
(i) There is a sequence of solid homogeneous order bounded sets (Bn)n
in X(µ) and a sequence of bounded homogeneous maps
Rn : X(µ) → E such that Rn ◦ φBn converges to T in the norm.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 25 / 37
53. Compactness and homogeneous maps on Banach function spaces
Domination of homogeneous maps
Theorem
∗ Let X(µ) be a Fatou order continuous Banach function space and let
T : X(µ) → E be a bounded homogeneous operator. Each of the
followings statements implies the next one.
(i) There is a sequence of solid homogeneous order bounded sets (Bn)n
in X(µ) and a sequence of bounded homogeneous maps
Rn : X(µ) → E such that Rn ◦ φBn converges to T in the norm.
(ii) There is a sequence of solid homogeneous sets Bn ⊆ X such that
their characteristic homogeneous operators are strongly dominated by
φp and a sequence of bounded homogeneous maps Rn : X(µ) → E
such that Rn ◦ φBn → T in the operator norm.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 25 / 37
54. Compactness and homogeneous maps on Banach function spaces
Domination of homogeneous maps
Theorem
∗ Let X(µ) be a Fatou order continuous Banach function space and let
T : X(µ) → E be a bounded homogeneous operator. Each of the
followings statements implies the next one.
(i) There is a sequence of solid homogeneous order bounded sets (Bn)n
in X(µ) and a sequence of bounded homogeneous maps
Rn : X(µ) → E such that Rn ◦ φBn converges to T in the norm.
(ii) There is a sequence of solid homogeneous sets Bn ⊆ X such that
their characteristic homogeneous operators are strongly dominated by
φp and a sequence of bounded homogeneous maps Rn : X(µ) → E
such that Rn ◦ φBn → T in the operator norm.
(iii) There is a sequence of homogeneous maps Pn : X → X that are
strongly dominated by φp and a sequence of bounded homogeneous
maps Rn : X(µ) → E such that Rn ◦ Pn converges to T in the norm.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 25 / 37
55. Compactness and homogeneous maps on Banach function spaces
Domination of homogeneous maps
(iv) T is the norm limit of a sequence Sn of homogeneous operators
factoring as
X(µ)
Sn
//
φn
E
X(µ)[1/p]
gn
// X(µ),
Rn
OO
where Rn are bounded homogeneous maps, gn ∈ X[1/p ] and
|φn| ≤ |φp|.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 26 / 37
56. Compactness and homogeneous maps on Banach function spaces
Domination of homogeneous maps
(iv) T is the norm limit of a sequence Sn of homogeneous operators
factoring as
X(µ)
Sn
//
φn
E
X(µ)[1/p]
gn
// X(µ),
Rn
OO
where Rn are bounded homogeneous maps, gn ∈ X[1/p ] and
|φn| ≤ |φp|.
(v) There is a sequence of homogeneous operators Sn so that there are
functions gn ∈ X[1/p ] such that
Sn(f ) ≤ f
1/p
X · gnf 1/p
X , f ∈ X, and Sn → T in the operator
norm.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 26 / 37
57. Compactness and homogeneous maps on Banach function spaces
Domination of homogeneous maps
(iv) T is the norm limit of a sequence Sn of homogeneous operators
factoring as
X(µ)
Sn
//
φn
E
X(µ)[1/p]
gn
// X(µ),
Rn
OO
where Rn are bounded homogeneous maps, gn ∈ X[1/p ] and
|φn| ≤ |φp|.
(v) There is a sequence of homogeneous operators Sn so that there are
functions gn ∈ X[1/p ] such that
Sn(f ) ≤ f
1/p
X · gnf 1/p
X , f ∈ X, and Sn → T in the operator
norm.
(vi) The operator T satisfies limµ(A)→0 supf ∈BX
T(f χA) = 0.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 26 / 37
58. Compactness and homogeneous maps on Banach function spaces
Compact linear operators
1 Basic definitions
2 Relevant subsets and maps
3 Domination of homogeneous maps
4 Compact linear operators
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 27 / 37
59. Compactness and homogeneous maps on Banach function spaces
Compact linear operators
Theorem
Let X(µ) be an order continuous Banach function space. An operator
T : X(µ) → E is compact if and only if it is essentially compact and there
is an order continuous Banach function space Z(ν) containing X such that
T extends to Z and a sequence of uniformly ν-absolutely continuous
homogeneous maps Qn : X → Z satisfying that T is the norm limit of the
sequence (T ◦ Qn)n.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 28 / 37
60. Compactness and homogeneous maps on Banach function spaces
Compact linear operators
Lemma
Let 1 p ∞. Let X(µ) be an order continuous Banach function space
and E be a Banach space. A continuous operator T : X(µ) → E is
essentially compact if and only if for every h ∈ X[1/p ] the map
Th : X[1/p] → E given by Th(·) := T(h ·), is compact.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 29 / 37
61. Compactness and homogeneous maps on Banach function spaces
Compact linear operators
Theorem
Let 1 ≤ p ∞. Let X(µ) be an order continuous Banach function space
and let E be a Banach space. The following statements for a continuous
operator T : X → E are equivalent:
(i) T is compact.
(ii) T is essentially compact and for every ε 0 there exists gε ∈ X[1/p ]
such that T(BX ) ⊂ T(gεBX[1/p]
) + εBE .
(iii) T is essentially compact and for every ε 0 there exists Kε 0 such
that T(BX ) ⊂ T(KεBX[1/p]
) + εBE .
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 30 / 37
62. Compactness and homogeneous maps on Banach function spaces
Compact linear operators
Corollary
Let 1 p ∞. Let X(µ) be an order continuous Banach function space,
E be a Banach space, and T : X(µ) → E be a essentially compact
continuous linear operator. Let Φ : BX → BX[1/p]
be a function. If
lim
K→∞
sup
f ∈BX
T(f χ{|f |≥K|Φ(f )|}) = 0
then T is compact.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 31 / 37
63. Compactness and homogeneous maps on Banach function spaces
Compact linear operators
Corollary
Let T be an essentially compact (positive) kernel operator
T : X(µ) → Y (ν) such that the kernel k satisfies that
lim
µ(A)→0
χAk(x, y) X (y)
Y
= 0.
Then T is compact.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 32 / 37
64. Compactness and homogeneous maps on Banach function spaces
Compact linear operators
We say that an operator T : X(µ) → E is µ-determined if T(χA) = 0
implies µ(A) = 0 for all A ∈ Σ.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 33 / 37
65. Compactness and homogeneous maps on Banach function spaces
Compact linear operators
We say that an operator T : X(µ) → E is µ-determined if T(χA) = 0
implies µ(A) = 0 for all A ∈ Σ.
Theorem
Let X(µ) be an order continuous Banach function space, E be a Banach
space and T : X → E be a µ-determined operator. If T is essentially
compact and satisfies
lim
µ(A)→0
sup
f ∈BX
T(f χA) = 0,
then T is compact.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 33 / 37
66. Compactness and homogeneous maps on Banach function spaces
Compact linear operators
Corollary
For a µ-determined operator, each one of the assertions in Theorem ∗
implies that T is compact.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 34 / 37
67. Compactness and homogeneous maps on Banach function spaces
Compact linear operators
Theorem
Let E be a Banach space, X(µ) be an order continuous Banach function
space with the Fatou property and 1 p ∞. The following statements
for a linear operator T : X(µ) → E are equivalent.
(i) T is compact.
(ii) T is essentially compact and there are gn ∈ X[1/p ], order bounded
homogeneous sets Bn and bounded homogeneous maps
φn : X → X[1/p] such that φn(BX ) ⊂ Bn and the sequence
T ◦ (gn · φn) converges to T in the norm.
(iii) T is essentially compact and there are order bounded homogeneous
sets Bn and bounded homogeneous maps φn : X → X such that
φn(BX ) ⊂ Bn and the sequence T ◦ φn converges to T in the norm.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 35 / 37
68. Compactness and homogeneous maps on Banach function spaces
Compact linear operators
(iv) T is essentially compact and there are order bounded homogeneous
sets Bn and bounded homogeneous maps φn : X → X such that the
sequence T ◦ φBn ◦ φn converges to T in the norm.
(v) T is essentially compact and there are bounded homogeneous maps
φn : X → X and homogeneous maps φp
n that are strongly dominated
by φp such that the sequence T ◦ φp
n ◦ φn converges to T in the norm.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 36 / 37
69. Compactness and homogeneous maps on Banach function spaces
Compact linear operators
Corollary
Let E be a Banach space, X(µ) be an order continuous Banach function
space with the Fatou property and 1 p ∞. The following statements
for a linear operator T : X(µ) → E are equivalent.
(i) T is compact.
(ii) T is essentially compact and there are order bounded homogeneous
sets Bn, bounded homogeneous maps φn : X → X and uniformly
µ-absolutely continuous homogeneous maps Rn : X → E such that
the sequence Rn ◦ φBn ◦ φn converges to T in the norm.
Enrique A. S´anchez P´erez (U.P. Valencia) Compactness and homogeneous maps on Banach function spaces 37 / 37