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. doi:10.7907/k7nd-5671. https://resolver.caltech.edu/CaltechETD:etd-09052007-110700

Abstract

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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.

Thesis Files