Citation
Jackson, Frances Yvonne (1998) SunDual Characterizations of the Translation Group of ℝ. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/3y5xkg66. https://resolver.caltech.edu/CaltechTHESIS:11212019103601328
Abstract
Let E be a Banach space. The mapping t → T (t) of ℝ (real numbers) into L_{b}(E), the Banach algebra of all bounded linear operators on E, is called a strongly continuous group or a C₀group, if G = {T(t) : t ∈ ℝ} defines a group representation of (ℝ, +) into the multiplicative group of L_{b}(E), and if ∀f ∈ E,
[equation; see abstract in scanned thesis for details].
For example, if E = C₀(ℝ), the function space which consists of all continuous, complex functions that vanish at infinity, then (∀t ∈ ℝ) (∀f ∈ C₀(ℝ)), the function T(t)f(x) = f(x + t), x ∈ ℝ, defines a strongly continuous group, since each f ∈ E is uniformly continuous; this group is called the translation group. If we now consider E = B(ℝ), the space of bounded, continuous complex functions on ℝ, then although the translation group on E is not strongly continuous, it is strongly continuous on the subspace BUC(ℝ) of E, which consists of bounded, uniformly continuous functions. BUC(ℝ) is the largest subspace of E on which the translation group is strongly continuous.
The adjoint family of a C₀group defined on a Banach space E, need not be strongly continuous on the Banach dual E* of E. Let E^{⊙} (pronounced Esun) be the largest linear subspace of E* relative to which the adjoint family is a C₀group:
[equation; see abstract in scanned thesis for details].
E^{⊙} is called the sundual or sunspace of E. If E = C₀(ℝ), then it follows from a wellknown result of A. Plessner that E^{⊙} = L¹(ℝ) ([Ple]). This research paper contains a characterization of the sundual of BUC(ℝ) and of the subspace W AP(ℝ) of BUC(ℝ), which consists of weakly almost periodic functions on ℝ.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  Mathematics  
Degree Grantor:  California Institute of Technology  
Division:  Physics, Mathematics and Astronomy  
Major Option:  Mathematics  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  11 June 1997  
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Record Number:  CaltechTHESIS:11212019103601328  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:11212019103601328  
DOI:  10.7907/3y5xkg66  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  13588  
Collection:  CaltechTHESIS  
Deposited By:  Mel Ray  
Deposited On:  21 Nov 2019 20:57  
Last Modified:  16 Apr 2021 22:12 
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