Evolution equations and semigroups of operators with the disjoint support property

Author: Biyanov, Andrey Y.

Year: 1995

Degree: Dissertation (Ph.D.)

Advisor: Luxemburg, W. A. J.

Committee Member: Unknown, Unknown

Option: Mathematics

DOI: 10.7907/k7nd-5671

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 ... 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 ... 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 ... 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 ... 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.

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