Files in this item
Files  Description  Format 

application/pdf 8600315.pdf (2MB)  (no description provided) 
Description
Title:  Semigroups of Composition Operators and the Cesaro Operator on H('p)(d) (Bergman Space, Infinitesimal Generator) 
Author(s):  Siskakis, Aristomenis Georgios 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(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 DenjoyWolff point of (phi)(,t) is in , we show that the condition (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) implies that the resolvent function R((lamda),(GAMMA)(,p)) is a compact operator on H('p). Here (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) In the absence of this condition an example shows that the spectrum of (GAMMA)(,p) can contain a halfplane 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. 
Issue Date:  1985 
Type:  Text 
Description:  82 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1985. 
URI:  http://hdl.handle.net/2142/71239 
Other Identifier(s):  (UMI)AAI8600315 
Date Available in IDEALS:  20141216 
Date Deposited:  1985 
This item appears in the following Collection(s)

Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois