Convex approximation properties of Banach spaces
Date
2011-05-16
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Abstract
Aproksimatsiooniprobleemiks nimetati üht kuulsaimat probleemi funktsionaalanalüüsi valdkonnas: kas igat kompaktset operaatorit Banachi ruumide vahel saab lähendada lõplikumõõtmeliste operaatoritega? Aastal 1955 näitas Grothendieck, et samaväärselt võib küsida, kas igal Banachi ruumil on aproksimatsiooniomadus. Aastal 1972 lahendas Enflo aproksimatsiooniprobleemi negatiivselt. Sellega seoses tekkis vajadus uurida aproksimatsiooniomaduse erinevaid versioone, mille kohta on püstitatud mitmeid olulisi lahendamata probleeme. Käesoleva väitekirja põhieesmärk on välja arendada ühtne meetod aproksimatsiooniomaduse erisuguste versioonide käsitlemiseks. Selliste versioonide hulka kuuluvad näiteks kompaktne aproksimatsiooniomadus, operaatorideaali poolt tekitatud aproksimatsiooniomadus ja Banachi võrede positiivne aproksimatsiooniomadus. Väitekirja põhiline mõiste on kumer aproksimatsiooniomadus ehk, täpsemalt, Banachi ruumil tegutsevate pidevate lineaarsete operaatorite nulli sisaldava kumera hulga poolt defineeritud aproksimatsiooniomadus. Väitekirjas saadud tulemused üldistavad Banachi ruumi ja tema kaasruumi klassikalise aproksimatsiooniomaduse kirjeldust kompaktsete operaatorite lähendamise kaudu. Samuti esitatakse lineaarse alamruumi poolt defineeritud aproksimatsiooniomaduse kirjeldus nõrgalt kompaktsete operaatorite lähendamise kaudu. Vaadeldakse tugeva aproksimatsiooniomaduse ja nõrgalt tõkestatud aproksimatsiooniomaduse mõisteid, mis olid sisse toodud Oja ja Lima poolt. Nende mõistete kumerate versioonide käsitlemiseks arendatakse välja ühtne lähenemismeetod. Üldistatakse Johnsoni teoreem meetrilise aproksimatsiooniomaduse ülekandumisest Banachi ruumilt tema kaasruumile. Väitekirjas loodud kumerate aproksimatsiooniomaduste teooriat rakendatakse Banachi võrede positiivse aproksimatsiooniomaduse ja Banachi ruumide paari poolt defineeritud aproksimatsiooniomaduse uurimisel.
The approximation problem was a long-standing open problem in the field of functional analysis. It asked whether every compact operator between Banach spaces could be approximated by finite-rank operators. In 1955 Grothendieck established its equivalence to the question whether every Banach space has the approximation property. In 1972 Enflo solved the question in the negative. This showed the necessity for investigating other variants of approximation properties. Nowadays, this area contains a number of important unsolved problems. The main aim of the thesis is to develop a unified approach to diverse versions of the approximation property, such as the compact approximation property, approximation properties defined by an operator ideal, or the positive approximation property of Banach lattices. Our principal notion is the approximation property defined by a convex set of operators containing 0, the convex approximation property. It turns out that this concept admits good counterparts of some important results on the classical approximation property. In the thesis we extend Grothendieck's description of the approximation property of a Banach space and its dual space via approximability of compact operators by finite-rank operators.We also describe the approximation property defined by a linear subspace of operators via the approximability of weakly compact operators. Then we look at the strong approximation property and the weak bounded approximation property of Oja and Lima, and we develop a unified approach to the treatment of their convex versions. We extend the famous Johnson’s lifting theorem, which permits to lift the metric approximation property to the dual space. Finally, we apply our theory of convex approximation properties to the positive approximation property of Banach lattices and to the approximation property defined for pairs of Banach spaces.
The approximation problem was a long-standing open problem in the field of functional analysis. It asked whether every compact operator between Banach spaces could be approximated by finite-rank operators. In 1955 Grothendieck established its equivalence to the question whether every Banach space has the approximation property. In 1972 Enflo solved the question in the negative. This showed the necessity for investigating other variants of approximation properties. Nowadays, this area contains a number of important unsolved problems. The main aim of the thesis is to develop a unified approach to diverse versions of the approximation property, such as the compact approximation property, approximation properties defined by an operator ideal, or the positive approximation property of Banach lattices. Our principal notion is the approximation property defined by a convex set of operators containing 0, the convex approximation property. It turns out that this concept admits good counterparts of some important results on the classical approximation property. In the thesis we extend Grothendieck's description of the approximation property of a Banach space and its dual space via approximability of compact operators by finite-rank operators.We also describe the approximation property defined by a linear subspace of operators via the approximability of weakly compact operators. Then we look at the strong approximation property and the weak bounded approximation property of Oja and Lima, and we develop a unified approach to the treatment of their convex versions. We extend the famous Johnson’s lifting theorem, which permits to lift the metric approximation property to the dual space. Finally, we apply our theory of convex approximation properties to the positive approximation property of Banach lattices and to the approximation property defined for pairs of Banach spaces.
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Keywords
Banachi ruumid, lähendamine, kumerad hulgad, dissertatsioonid, Banach spaces, approximation, convex sets