University of Illinois Urbana-Champaign

Semigroups of Composition Operators and the Cesaro Operator on H('p)(d) (Bergman Space, Infinitesimal Generator)

Siskakis, Aristomenis Georgios

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Description

Title
Semigroups of Composition Operators and the Cesaro Operator on H('p)(d) (Bergman Space, Infinitesimal Generator)
Author(s)
Siskakis, Aristomenis Georgios
Issue Date
1985
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Abstract
  • A semigroup T(,t) : t (GREATERTHEQ) 0 of composition operators on H('P)( ) arises as T(,t)(f) = f(CCIRC)(phi)(,t) where (phi)(,t) : t (GREATERTHEQ) 0 is a semigroup of analytic functions mapping the unit disk into itself. The infinitesimal
  • generator (GAMMA)(,p) of T(,t) is given by (GAMMA)(,p)(f) = Gf' where G is the infinites- imal generator of
  • on H('P) is equal to p for 2 (LESSTHEQ) p < (INFIN) and is between p and 2 for 1 (LESSTHEQ) p < 2. Also the spectrum of C is shown to be z : (VBAR)z - P/2(VBAR) (LESSTHEQ) P/2 for 2 (LESSTHEQ) p < (INFIN) and to contain this set if 1 (LESSTHEQ) p < 2. Similar results are proved for an averaging operator A related to C.
  • (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)
  • There is a univalent analytic function h : (--->) (//C) associated with each semigroup of functions (phi)(,t) , defined as the solution of a certain functional equation involving (phi)(,t) .
  • In this work we investigate the relation between the functional analytic properties of the (unbounded) operator (GAMMA)(,p) and the univalent
  • function h. The point spectrum of (GAMMA)(,p) is characterized in terms of h. If the Denjoy-Wolff point of (phi)(,t) is in , we show that the condition
  • implies that the resolvent function R((lamda),(GAMMA)(,p)) is a compact operator on H('p). Here
  • In the absence of this condition an example shows that the spectrum of (GAMMA)(,p) can contain a half-plane so R((lamda),(GAMMA)(,p)) need not always be com- pact. Although this condition is shown to be satisfied frequently, an example shows that it is not necessary for compactness.
Graduation Semester
1985
Type of Resource
text
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