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Taspaganbetova, Zhanar (2013) Boundedness and compactness of matrix operators in weighted spaces of sequences and their applications. [Ph.D. thesis]

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Abstract (english)

The present Thesis is dedicated to the investigation of necessary and sufficient conditions for which a weighted Hardy type inequality holds in weighted spaces of sequences and on the cone of non-negative monotone sequences, and their applications.
We prove a new discrete Hardy type inequality involving a kernel which has a more general form than those known in the literature.

This Thesis consists of four chapters.

In Chapter 1, we shortly describe the development and current status of the theory of Hardy type inequalities. Moreover, Chapter 1 includes the statement and motivation of the problems and the main results. In Chapter 1, we also present
some well-known auxiliary facts and necessary notation on Hardy type inequalities in weighted spaces of sequences and on the cone of non-negative monotone sequences.

In Chapter 2, we study the problems of boundedness and compactness of matrix operators in weighted spaces of equences. We introduce a general class of matrices, and introduce their properties. Moreover, Chapter 2 contains examples of matrices from the introduced classes and here we show that such classes of matrices include well-known classical operators such as the operator of multiple summation, Hoelder's operator, Cesaro operator and others.
We establish necessary and sufficient conditions for the boundedness and compactness of the matrix operators in weighted spaces of sequences, where the corresponding matrices belong to such classes. Such classes of matrices are wider than those which have been previously studied in the theory of discrete Hardy type inequalities. Moreover, some related results are also proved.

In Chapter 3, we investigate a Hardy type inequality restricted
to the cone of non-negative and non-increasing sequences under weaker conditions than those studied before in the literature. We obtain new results, which generalize the known results concerning this subject.

Chapter 4 is devoted to the application of the main results.
Here we apply the main results of Chapter 2 in order to obtain criteria on boundedness and compactness of composition of matrix operators in weighted spaces of sequences.
By using the results of Chapter 2 we obtain necessary and sufficient conditions for which three-weighted Hardy type inequalities hold. Moreover, in Chapter 4, by exploiting the main results of Chapter 2 and 3 we obtain two-sided estimates for summable matrices in weighted spaces of sequences and on the cone of non-negative and non-increasing sequences.

Abstract (italian)

Questa tesi è dedicata allo studio di condizioni necessarie e sufficienti per cui valga una disuguaglianza di tipo Hardy con peso in uno spazio pesato di successioni e nel cono delle successioni monotone non-negative, e alle corrispondenti applicazioni.
Proviamo una nuova disuguaglianza discreta di tipo Hardy con nucleo di forma più generale di quelli noti in letteratura.

La tesi consiste di quattro capitoli.

Nel Capitolo 1 descriviamo brevemente lo sviluppo e lo stato attuale della teoria delle disuguaglianze di tipo Hardy. Inoltre il Capitolo 1 contiene l'enunciato e la motivazione dei problemi e dei principali risultati. Nel Capitolo 1 presentiamo anche alcuni fatti ausiliari ben noti e la notazione necessaria per le disuguaglianze di tipo Hardy negli spazi pesati di successioni e nel cono delle successioni monotone non-negative.

Nel Capitolo 2 studiamo il problema della limitatezza e compattezza degli operatori matriciali negli spazi pesati di uccessioni. Introduciamo una classe generale di matrici e le loro proprietà. Inoltre il Capitolo 2 contiene esempi di matrici delle classi introdotte e qui mostriamo che tali classi di matrici contengono ben noti operatori classici come l'operatore di sommazione multipla, l'operatore di Hoelder, l'operatore di Cesaro ed altri. Stabiliamo condizioni necessarie e sufficienti per la limitatezza e la compattezza di operatori matriciali in spazi pesati di successioni, nel caso in cui le corrispondenti matrici appartengano a tali classi.
Tali classi di matrici sono più grandi di quelle che sono state studiate in precedenza nella teoria delle disuguaglianza discrete di tipo Hardy. Inoltre, si dimostrano anche dei risultati ad esse relativi.

Nel Capitolo 3, studiamo una disuguaglianza di tipo Hardy ristretta al cono delle successioni non-negative e non crescenti in condizioni più deboli di quelle studiate prima nella letteratura. Otteniamo dei nuovi risultati che generalizzano i risultati noti su questo argomento.

Il Capitolo 4 è dedicato alle applicazioni dei risultati principali. Qui applichiamo i risultati principali del Capitolo 2 al fine di ottenere criteri di limitatezza e compattezza per la composizione di operatori matriciali in spazi pesati di successioni. Utilizzando i risultati del Capitolo 2 otteniamo condizioni necessarie e sufficienti affinchè valgano disuguaglianze di tipo Hardy a tre pesi. Inoltre nel Capitolo 4, sfruttando i risultati dei Capitoli 2 e 3 otteniamo stime bilatere per matrici sommabili in spazi pesati di successioni e nel cono delle successioni non negative e non crescenti.

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EPrint type:Ph.D. thesis
Tutor:Taspaganbetova, Zhanar
Ph.D. course:Ciclo 25 > Scuole 25 > SCIENZE MATEMATICHE > MATEMATICA
Data di deposito della tesi:24 April 2013
Anno di Pubblicazione:30 April 2013
Key Words:weighted inequalities; discrete Hardy-type inequalities; matrix operators; boundedness; compactness; monotone sequences; summation methods; compositions of matrix operators; three-weighted inequality of Hardy type
Settori scientifico-disciplinari MIUR:Area 01 - Scienze matematiche e informatiche > MAT/05 Analisi matematica
Struttura di riferimento:Dipartimenti > Dipartimento di Matematica
Codice ID:6075
Depositato il:14 Oct 2013 10:59
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