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Factorization and Domination Properties of Operators on Banach Lattices UNIVERSIDAD COMPLUTENSE DE MADRID FACULTAD DE CIENCIAS MATEMÁTICAS Departamento de Análisis Matemático FACTORIZATION AND DOMINATION PROPERTIES OF OPERATORS ON BANACH LATTICES. MEMORIA PARA OPTAR AL GRADO DE DOCTOR PRESENTADA POR Pedro Tradacete Pérez Bajo la dirección de los doctores Francisco L. Hernández Rodríguez Julio Flores Álvarez Madrid, 2010 ISBN: 978-84-693-7735-2 © Pedro Tradacete Pérez, 2009 Factorization and domination properties of operators on Banach lattices Factorizaci´on,mayoraci´ony potencias de operadores positivos en ret´ıculosde Banach Pedro Tradacete P´erez Directores de Tesis: Francisco L. Hern´andezRodr´ıguez Julio Flores Alvarez´ Memoria para optar al grado de Doctor en Matem´aticas. Febrero de 2009 Departamento de An´alisisMatem´atico. Universidad Complutense de Madrid. Contents Introducci´on,objetivos y descripci´onde resultados 5 Introduction 17 Chapter 1. Preliminary results 27 1. Basic facts 27 2. Order continuous Banach lattices 28 3. Convexity, concavity, and related notions 30 4. Equi-integrable sets in Banach lattices 32 5. Operators on Banach lattices 34 6. Some Banach spaces results and notation 38 Chapter 2. Power and domination problems for strictly singular operators 41 1. Domination results 41 2. Powers of dominated operators 46 3. Related questions 49 Chapter 3. A Characterization of strictly singular operators in Banach lattices 53 1. Relation with `2-singular and DSS operators 53 2. The role of the lower 2-estimate 58 Chapter 4. Invariant subspaces of positive strictly singular operators 67 1. R-condition 68 2. Strictly singular and AM-compact operators 70 3. Invariant subspaces of positive strictly singular operators 71 4. Invariant subspaces of SS-friendly operators 74 Chapter 5. Compact products of strictly singular operators 77 1. Disjointly homogeneous Banach lattices 77 2. Compact squares of strictly singular operators 79 3. Strictly singular operators on Lorentz spaces 83 4. Strictly singular operators on Orlicz spaces 88 5. Duality of strictly singular operators in spaces Lp;q 91 Chapter 6. c0-singular and `1-singular operators between vector valued Banach lattices 97 1. Tools 97 3 4 CONTENTS 2. Main results 102 3. Applications 103 Chapter 7. Factorization and domination of Banach-Saks operators 107 1. Factorization of operators through Banach lattices with the Banach-Saks property 107 2. Domination by Banach-Saks operators 112 3. Convex hulls of Banach-Saks sets 114 Chapter 8. Factorization of p-convex and q-concave operators 117 1. The basic construction 117 2. Duality relations 125 3. Complex interpolation of Banach lattices 133 4. Factorization for operators which are both p-convex and q-concave 139 5. Other results 146 Bibliography 149 Introducci´on,objetivos y descripci´onde resultados El tema central de esta Memoria es el estudio de operadores entre ret´ıculosde Banach. Recordemos que un ret´ıculode Banach es un espacio de Banach E dotado de una norma k · k, y un orden parcial ≤ tales que: • x ≤ y implica x + z ≤ y + z, para cualesquiera x; y; z 2 E, • λx ≥ 0, para todo x ≥ 0 en E y todo n´umeroreal λ ≥ 0, • para cualesquiera x; y 2 E existe un m´ınimoy un supremo de x e y en E, denotados respectivamente x ^ y, x _ y, • si denotamos jxj = x _ (−x), que verifica jxj 2 E por la propiedad anterior, se tiene que kxk ≤ kyk si jxj ≤ jyj. Es decir, en un ret´ıculode Banach la estructura lineal, reticular y la topolog´ıason com- patibles, y el estudio de relaciones entre estas tres estructuras es uno de los objetivos de esta teor´ıa.Por una parte, en un ret´ıculode Banach la estructura adicional de ret´ıculoproporciona gran cantidad de herramientas de las que no disponemos en espacios de Banach m´asgenerales. Este hecho facilita el estudio de propiedades geom´etricasde ret´ıculosde Banach, as´ıcomo de las propiedades de operadores que act´uanentre ret´ıculos. Por otra parte, la relaci´onde orden en un ret´ıculode Banach induce tambi´enun orden en los espacio de operadores que actuan entre dos ret´ıculos. Este orden se define de manera nat- ural, considerando un operador positivo como aqu´elque env´ıaelementos positivos a elementos positivos, y definiendo para dos operadores R y T , R ≤ T si la diferencia T − R es un operador positivo. A partir de este hecho, surgen de manera natural una serie de cuestiones sobre la relaci´onentre las propiedades de los operadores y la estructura reticular. Entre estas preguntas, una de las m´asfrecuentes es el llamado problema de mayoraci´on, que consiste en dados oper- adores S ≤ T entre ret´ıculosde Banach E y F , estudiar qu´epropiedades hereda S del operador T . Concretamente, dada una familia de operadores I, por ejemplo un ideal en el sentido de Pietsch, se considerar´abajo qu´econdiciones en los ret´ıculos E y F , si 0 ≤ R ≤ T : E ! F y T 2 I, entonces R 2 I. En este sentido tenemos resultados cl´asicosde P. G. Dodds y D. H. Fremlin, quienes de- mostraron en [37] para ret´ıculos E y F , con E∗ y F orden continuos, si 0 ≤ R ≤ T : E ! F y T es compacto, entonces R tambi´enes compacto. En la misma l´ınea,A. W. Wickstead ha mostrado en [136] que si E∗ o F son ret´ıculosorden continuos y T es d´ebilmente compacto, entonces R tambi´enes d´ebilmente compacto. A su vez, N. Kalton y P. Saab han demostrado en [77] que si F es orden continuo y T es Dunford-Pettis, entonces R es tambi´enDunford-Pettis. 5 6 INTRODUCCION,´ OBJETIVOS Y DESCRIPCION´ DE RESULTADOS M´asrecientemente, C. C. A. Labuschagne ha obtenido tambi´enresultados en esta l´ıneapara operadores Asplund y Radon-Nikodym (ver [83]). Los origenes del teorema de Dodds-Fremlin, y de los problemas de mayoraci´on,est´aen los trabajos de W. A. Luxemburg y A. C. Zaanen sobre operadores integrales [92]. El teorema de Dodds-Fremlin fue demostrado previamente para operadores integrales por R. J. Nagel y U. Schlotterbeck en [97], independientemente de las versiones existentes en la literatura sovi´etica (ver [79, Theorem 5.10]). La validez de dicho teorema para operadores arbitrarios en espacios Lp fue conjeturada a partir de evidencias f´ısicaspor los f´ısicosmatem´aticosJ. Avron, I. Herbst y B. Simon en [16], y demostrada por L. D. Pitt en [109]. Practicamente al mismo tiempo apareci´ola demostraci´ongeneral de P. G. Dodds y D. H. Fremlin que ha sido la m´asinfluyente en la teor´ıaactual de operadores en ret´ıculosde Banach. En el caso particular de endomorfismos en un mismo ret´ıculo,es decir si E = F , una cuesti´on interesante relacionada con la anterior es el llamado problema de la potencia, que consiste en dados dos operadores 0 ≤ R ≤ T : E ! E saber si alguna potencia de R hereda propiedades de T , en este caso sin pedir condiciones extra al ret´ıculo E. Esta pregunta fue inicialmente planteada por C. D. Aliprantis y O. Burkinshaw en [9] y [10], donde obtuvieron para cualquier ret´ıculode Banach E y operadores 0 ≤ R ≤ T : E ! E, si T es compacto, entonces R3 es compacto, y si T es d´ebilmente compacto, entonces R2 es d´ebilmente compacto. En la misma l´ınea,para la clase de operadores Dunford-Pettis, N. Kalton y P. Saab, probaron que dado un ret´ıculode Banach E, si 0 ≤ R ≤ T : E ! E y T es Dunford-Pettis, entonces R2 tambi´enes Dunford-Pettis. Adem´as,estos resultados son ´optimos,en el sentido de que existen ejemplos de operadores que no cumplen estas propiedades para potencias menores. En la actualidad, podemos encontrar aplicaciones de estos resultados en an´alisis de modelos de biolog´ıamatem´aticao en aplicaciones de la ecuaci´ondel transporte. Concretamente, en biolog´ıamatem´atica,la primera aplicaci´onde estos resultados se debe a R. Burger en [27], que ha utilizado los resultados de [9] para obtener informaci´onsobre los autovalores de operadores involucrados en un determinado modelo discreto que describe la evoluci´onde densidades en una poblaci´onde reproducci´onasexual bajo la acci´onde mutaci´ony selecci´on. En una l´ınea similar, A. Rhandi y R. Schnaubelt en [118] han utilizado los resultados de mayoraci´onpara obtener la existencia y unicidad de soluciones positivas de la ecuaci´onque describe un modelo de poblaciones no autonomas con difusi´onen L1. Por otro lado, en el estudio de la ecuaci´on del transporte, estos resultados han sido aplicados recientemente por A. Dehici, A. Jeribi y K. Latrach en [35] para estudiar el espectro de operadores de transici´onque describen la madurez de una poblaci´on creciente de c´elulas,y por M. Sbihi en [121] para estudiar la estabilidad del espectro esencial de semigrupos en un espacio de Hilbert. Estas´ son s´oloalgunas de las aplicaciones que pretendemos ilustren la utilidad de estos resultados en otras ´areasde las matem´aticas,vease [2] para m´asinformaci´on. En esta memoria, dos cap´ıtulos de la misma giran en torno a estos problemas. En concreto, en el Cap´ıtulo2, estudiamos el problema de mayoraci´ony de la potencia para operadores INTRODUCCION,´ OBJETIVOS Y DESCRIPCION´ DE RESULTADOS 7 estrictamente singulares. Recordemos que un operador T : X ! Y entre espacios de Banach es estrictamente singular (o m´asbrevemente SS) si para ning´unsubespacio (cerrado) M ⊂ X de dimensi´oninfinita, la restricci´on T jM es invertible. Esta clase de operadores fue introducida por T. Kato en [78], en conexi´oncon la teor´ıade perturbaci´onde operadores Fredholm. Los operadores estrictamente singulares forman un ideal cerrado de operadores (en el sentido de Pietsch), que contiene al ideal de los operadores compactos.
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