Generating systems of sets and sequences
Date
2017-06-28
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Abstract
Operaatorideaalide teooria sai alguse A. Pietschi monograafiast ning on tänaseks saanud kaasaegse Banachi ruumide teooria lahutamatuks osaks. I. Stephani tõi sisse kaks operaatorideaalidega tihedalt seotud mõistet: genereerivate hulkade süsteem ja genereerivate jadade süsteem. Nimelt, lähtudes kahest etteantud genereerivate hulkade süsteemist, saame me tekitada operaatorideaali, mis koosneb kõigist operaatoritest, mis teisendavad esimesse süsteemi kuuluvad hulgad teise süsteemi kuuluvateks hulkadeks. Genereerivate jadade süsteeme saab omakorda kasutada genereerivate hulkade süsteemide tekitamiseks.
Väitekirjas uuritakse genereerivate hulkade ja jadade süsteemide klasse ning nendevahelisi seoseid. Muuhulgas tõestatakse, et leidub Galois' vastavus genereerivate hulkade süsteemide klassi ja teatava genereerivate jadade süsteemide faktorklassi vahel. Lisaks vaadeldakse eelmainitud struktuure ning nendega seotud klasse võreteoreetilisest aspektist.
Üks levinud näide genereerivate hulkade süsteemide kohta on kõigi suhteliselt kompaktsete hulkade süsteem. A. Grothendieck tõestas 1955. aastal, et Banachi ruumi alamhulk on suhteliselt kompaktne parajasti siis, kui ta sisaldub nulli koonduva jada kinnises kumeras kattes. Väitekirjas uuritakse mitmeid kirjanduses varasemalt sisse toodud alternatiivseid suhtelise kompaktsuse mõisteid, mis baseeruvad sellel tulemusel.
Väitekirjas tuuakse sisse üldine meetod, mis tekitab etteantud normeeritud jadaruumist ja normeeritud jadade süsteemist operaatorideaali ja varustab selle teatava kvaasinormiga. Tõestatakse, et sobivatel eeldustel on tulemuseks kvaasi-Banachi operaatorideaal. Selle konstruktsiooni näidetena saadakse uusi tulemusi eelmainitud alternatiivsete suhtelise kompaktsuse mõistete kohta.
A. Pietsch created the theory of operator ideals, which has been widely adopted and permeates the contemporary field of Banach spaces. I. Stephani introduced the related notions of a generating system of sets and a generating system of sequences. Namely, given two generating systems of sets, one obtains an operator ideal by considering all of the operators that map the sets of the first system to the sets of the second system. Generating systems of sequences can be used to obtain generating systems of sets. This thesis studies the classes of generating systems of sets and sequences and the relations between them; in particular, we show that there is a Galois connection between the former and a certain quotient class of the latter. We also study the lattice structure of various classes of operator ideals, generating systems of sets, and generating systems of sequences. A well-known example of a generating system of sets is the system of relatively compact sets. A. Grothendieck proved in 1955 that a subset of a Banach space is relatively compact if and only if it is contained in the closed convex hull of a norm null sequence. Based on this result, various alternative notions of relatively compact sets have been introduced in the literature. Several of these notions are studied thoroughly in this thesis. We propose a general method for constructing generating systems of sets and operator ideals from a normed sequence space and a normed system of sequences. We prove that the constructed operator ideal is always quasi-Banach provided that certain assumptions are met. As examples of this construction, we obtain new results for some aforementioned alternative notions of relative compactness
A. Pietsch created the theory of operator ideals, which has been widely adopted and permeates the contemporary field of Banach spaces. I. Stephani introduced the related notions of a generating system of sets and a generating system of sequences. Namely, given two generating systems of sets, one obtains an operator ideal by considering all of the operators that map the sets of the first system to the sets of the second system. Generating systems of sequences can be used to obtain generating systems of sets. This thesis studies the classes of generating systems of sets and sequences and the relations between them; in particular, we show that there is a Galois connection between the former and a certain quotient class of the latter. We also study the lattice structure of various classes of operator ideals, generating systems of sets, and generating systems of sequences. A well-known example of a generating system of sets is the system of relatively compact sets. A. Grothendieck proved in 1955 that a subset of a Banach space is relatively compact if and only if it is contained in the closed convex hull of a norm null sequence. Based on this result, various alternative notions of relatively compact sets have been introduced in the literature. Several of these notions are studied thoroughly in this thesis. We propose a general method for constructing generating systems of sets and operator ideals from a normed sequence space and a normed system of sequences. We prove that the constructed operator ideal is always quasi-Banach provided that certain assumptions are met. As examples of this construction, we obtain new results for some aforementioned alternative notions of relative compactness
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funktsionaalanalüüs, Banachi ruumid, operaator-ideaalid, functional analysis, Banach spaces, operator ideals