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The Resource Weakly analytic vector-valued measures, by Annela Rämmer Kelly

Weakly analytic vector-valued measures, by Annela Rämmer Kelly

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Summary
The research for my dissertation focuses on studying vector-valued measures that are weakly analytic and scalar-valued measures that are Riesz-analytic. My research stems from a well-known result in harmonic analysis; the celebrated F. and M. Riesz theorem. The main result of the dissertation generalizes an extension of this result for vector-valued measures. For scalar-valued measures, the classical F. and M. Riesz theorem states that if $\mu,$ a Borel measure defined on the unit circle T, is analytic, then $\mu$ is absolutely continuous with respect to Lebesgue measure m. There are several extensions to this theorem. Lately, Asmar and Montgomery-Smith generalized the conclusion that $\mu$ translates continuously for measures on locally compact Hausdorff spaces. They proved that under certain conditions on T, where $T=\{T\sb{t}\}\sb{t\in R}$ is a collection of uniformly bounded invertible isomorphisms of the space of measures, whenever the measure $\mu$ is weakly analytic, then the mapping $t\to T\sb{t}\mu$ is continuous. We will prove a similar result for vector-valued measures. More precisely, let $\Sigma$ be the $\sigma$-algebra of subsets of a set $\Omega$ and $Y\sp{\*}$ be the dual of a Banach space Y. We will assume that $Y\sp{\*}$ has the ARNP and consider vector-valued measures taking values in $Y\sp{\*}.$ Let $T=\{T\sb{t}\}\sb{t\in R}$ be a collection of uniformly bounded invertible isomorphisms of this space of measures. A vector-valued measure $\mu$ with values in $Y\sp{\*}$ is called weakly analytic if $\forall A\in\Sigma,\ \forall y\in Y,$ the mapping $t\to x(T\sb{t}\mu(A))\in H\sp{\infty}(\IR).$ In the dissertation we will show that if a measure $\mu$ is weakly analytic, then under certain conditions on $T,\ t\to T\sb{t}\mu$ is absolutely continuous. Let us return to the scalar-valued measures again. DeLeeuw and Glicksberg generalized the F. and M. Riesz theorem, they showed that if a Borel measure on a compact abelian group G is $\varphi$-analytic, then the measure is quasi-invariant. We will follow their footsteps and show that if $\mu$ is Riesz-E-analytic and $E\subset\ G$ satisfies certain conditions, then $\mu$ is quasi-invariant
Language
eng
Work
Publication
Extent
v, 62 leaves
Note
  • Typescript
  • Vita
Instance
Label
Weakly analytic vector-valued measures
Title
Weakly analytic vector-valued measures
Statement of responsibility
by Annela Rämmer Kelly
Creator
Subject
Language
eng
Summary
The research for my dissertation focuses on studying vector-valued measures that are weakly analytic and scalar-valued measures that are Riesz-analytic. My research stems from a well-known result in harmonic analysis; the celebrated F. and M. Riesz theorem. The main result of the dissertation generalizes an extension of this result for vector-valued measures. For scalar-valued measures, the classical F. and M. Riesz theorem states that if $\mu,$ a Borel measure defined on the unit circle T, is analytic, then $\mu$ is absolutely continuous with respect to Lebesgue measure m. There are several extensions to this theorem. Lately, Asmar and Montgomery-Smith generalized the conclusion that $\mu$ translates continuously for measures on locally compact Hausdorff spaces. They proved that under certain conditions on T, where $T=\{T\sb{t}\}\sb{t\in R}$ is a collection of uniformly bounded invertible isomorphisms of the space of measures, whenever the measure $\mu$ is weakly analytic, then the mapping $t\to T\sb{t}\mu$ is continuous. We will prove a similar result for vector-valued measures. More precisely, let $\Sigma$ be the $\sigma$-algebra of subsets of a set $\Omega$ and $Y\sp{\*}$ be the dual of a Banach space Y. We will assume that $Y\sp{\*}$ has the ARNP and consider vector-valued measures taking values in $Y\sp{\*}.$ Let $T=\{T\sb{t}\}\sb{t\in R}$ be a collection of uniformly bounded invertible isomorphisms of this space of measures. A vector-valued measure $\mu$ with values in $Y\sp{\*}$ is called weakly analytic if $\forall A\in\Sigma,\ \forall y\in Y,$ the mapping $t\to x(T\sb{t}\mu(A))\in H\sp{\infty}(\IR).$ In the dissertation we will show that if a measure $\mu$ is weakly analytic, then under certain conditions on $T,\ t\to T\sb{t}\mu$ is absolutely continuous. Let us return to the scalar-valued measures again. DeLeeuw and Glicksberg generalized the F. and M. Riesz theorem, they showed that if a Borel measure on a compact abelian group G is $\varphi$-analytic, then the measure is quasi-invariant. We will follow their footsteps and show that if $\mu$ is Riesz-E-analytic and $E\subset\ G$ satisfies certain conditions, then $\mu$ is quasi-invariant
Additional physical form
Also available on the Internet.
Cataloging source
MUU
http://library.link/vocab/creatorDate
1967-
http://library.link/vocab/creatorName
Kelly, Annela Rämmer
Degree
Ph. D.
Dissertation year
1996.
Government publication
government publication of a state province territory dependency etc
Granting institution
University of Missouri-Columbia
Index
no index present
Literary form
non fiction
Nature of contents
bibliography
http://library.link/vocab/subjectName
  • Vector-valued measures
  • Riesz spaces
  • Hausdorff measures
Target audience
specialized
Label
Weakly analytic vector-valued measures, by Annela Rämmer Kelly
Instantiates
Publication
Note
  • Typescript
  • Vita
Bibliography note
Includes bibliographical references (leaves 60-61)
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier
Color
not applicable
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Control code
38597354
Dimensions
29 cm
Extent
v, 62 leaves
Media category
computer
Media MARC source
rdamedia
Media type code
  • c
Specific material designation
remote
Label
Weakly analytic vector-valued measures, by Annela Rämmer Kelly
Publication
Note
  • Typescript
  • Vita
Bibliography note
Includes bibliographical references (leaves 60-61)
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier
Color
not applicable
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Control code
38597354
Dimensions
29 cm
Extent
v, 62 leaves
Media category
computer
Media MARC source
rdamedia
Media type code
  • c
Specific material designation
remote

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