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Evolution equations and semigroups of operators with the disjoint support property

Citation

Biyanov, Andrey Y. (1995) Evolution equations and semigroups of operators with the disjoint support property. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-09052007-110700

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.

Let [...], [...] be locally compact Hausdorff spaces, [...], [...] Banach spaces.

Theorem. T is an operator in [...], [...] with the disjoint support property if and only if [...] open, [...] such that:

(1) [...]. (2) [...] compact, [...] compact, [...] with the following property: [...]. (3) [...]

[...].

Let X be a locally compact Hausdorff space, E a Banach space.

Theorem. [...] is a [...]-group on [...](X,E) with the disjoint support property if and only if [...] a continuous flow, [...] a continuous cocycle of [...] such that [...].

There is a corresponding result about [...]-semigroups on [...](X,E) with the disjoint support property, where semiflows and semicocycles play the roles of flows and cocycles respectively.

Suppose [...], X is either (a,b) or [a,b], where by [[...],b] we mean ([...],b], and by [a,[...]] we mean [a,[...]).

Theorem. Let [...] be a [...]-group on [...](X) with the disjoint support property. Then [...] is the union of pairwise disjoint intervals [...], [...], where I is either finite or countable and [...]: [...] such that [...] = [...] : [...] is a homeomorphism and the corresponding group dual

[...].

The above theorem generalizes the well-known result of A. Plessner that if [...] and [...], then f is absolutely continuous if and only if [...].

The following theorem generalizes the result of N. Wiener and R. C. Young about the behavior of measures on [...] under translation.

Theorem. Let [...] be a [...]-group on [...](X) with the disjoint support property. Then [...]

lim sup[...],

where [...] is the component of in [...]. Moreover, if lim sup[...] = 1, then the last inequality becomes an equality.

Item Type:Thesis (Dissertation (Ph.D.))
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Restricted to Caltech community only
Research Advisor(s):
  • Luxemburg, W. A. J.
Thesis Committee:
  • Unknown, Unknown
Defense Date:20 April 1995
Record Number:CaltechETD:etd-09052007-110700
Persistent URL:http://resolver.caltech.edu/CaltechETD:etd-09052007-110700
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:3339
Collection:CaltechTHESIS
Deposited By: Imported from ETD-db
Deposited On:11 Sep 2007
Last Modified:26 Dec 2012 02:59

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