Compactness and null sequences defined by lp spaces

Abstract

Funktsionaalanalüüsi ja selle rakenduste alusuuringutes mängib olulist rolli hulkade ja operaatorite kompaktsus. Grothendiecki kompaktsuse kriteeriumi kohaselt on Banachi ruumi alamhulk suhteliselt kompaktne parajasti siis, kui ta sisaldub mingi nulliks koonduva jada kinnises kumeras kattes. Sellest kriteeriumist inspireerituna on sisse toodud mitmeid uusi tugevamaid kompaktsuse versioone, mis omakorda defineerivad erinevaid kompaktsete operaatorite klasse. Käesolevas väitekirjas töötatakse välja kompaktsuse vormide ning vastavate operaatorite klasside ühtne teooria. Selleks tuuakse sisse tugevam ja üldisem (p,r)-kompaktsuse mõiste nii hulkade kui ka operaatorite jaoks, mis erijuhul p=∞ ühtib klassikalise kompaktsusega ning juhul r=1 osutub Bourgain-Reinovi p-kompaktsuseks. Erijuhul r= p/(p-1) on tegemist viimasel ajal intensiivselt uuritud Sinha-Karni p-kompaktsusega. Väitekirja tunduvalt lihtsam alternatiivne teooria hõlmab muuhulgas ka Sinha-Karni p-kompaktsust ning saadud tulemused parendavad mitmeid varasemaid.

The compactness of sets and operators plays an important role in the baseline studies of functional analysis and its applications. According to Grothendieck’s criterion of compactness, a subset of a Banach space is relatively compact if and only if it is contained in the closed convex hull of a sequence converging to zero. This criterion has inspired several other forms of relative compactness, which in turn define different classes of compact operators. In the current thesis, a unified study of compactnesses and corresponding classes of compact operators is developed. A stronger and more general notion of (p,r)-compactness for sets and for operators is introduced, so that when p=∞, it coincides with the classical compactness and for r=1, it is exactly the p-compactness in the sense of Bougain-Reinov. In the special case when r=p/(p-1), it is the extensively studied p-compactness in the sense of Sinha-Karn. The alternative theory of the thesis encompasses among others the p-compactness in the sense of Sinha-Karn, and the results improve several known ones.