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|Title:||Spectral Decompositions of Isometries on Hardy Spaces of the Disk and the Torus|
|Department / Program:||Mathematics|
|Degree Granting Institution:||University of Illinois at Urbana-Champaign|
|Abstract:||Spectral decompositions of strongly continuous one-parameter groups of surjective isometries on H('p)( ID ) and H('p)( TT ('2)), as well as logarithms of (single) members of these groups along with their spectral decompositions are examined, mainly in light of the abstract results derived by E. Berkson, T. A. Gillespie and P. S. Muhly (1986) concerning uniformly bounded one-parameter groups and (single) power-bounded operators on UMD spaces.
The pointwise actions of the above mentioned spectral decom- positions and logarithms on H('p)( ID ), 1 < p < +(INFIN), are concretely presented in the case of parabolic isometric groups and single parabolic isometries. Analogous results are presented for parabolic isometric groups on H('p)( TT ('2)) and single parabolic isometries on H('2) ( TT ('2)). In addition, a mechanism is introduced which can be used to achieve concrete descriptions of the pointwise action of spectral decompositions for hyperbolic isometric groups on H('p)( ID ), 1 < p < +(INFIN). An example of how this mechanism can be applied is also presented.
The above descriptions not only independently reproduce results achieved by E. Berkson (1985) for the spectral decompositions asso- ciated with parabolic isometric groups on H('p)( ID ), 1 < p < +(INFIN), but also illuminate the role that these projections play in the structure of Hardy spaces. Furthermore they provide us with a foundation so that we can derive new results, such as: a description of the action of the partial sum-operators of M. Riesz when carried from L('p)((//R)) to H('p)( ID ), 1 < p (LESSTHEQ) 2, the introduction of formulae that generate hard to detect Hardy class-functions, and the connection of the action of spectral projections associated with hyperbolic isometric groups with hypergeometric functions.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1986.
|Date Available in IDEALS:||2014-12-16|