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 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 Urbana-Champaign 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 infinitesimalgenerator (GAMMA)(,p) of T(,t) is given by (GAMMA)(,p)(f) = Gf' where G is the infinites- imal generator ofon 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 univalentfunction 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(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 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. Issue Date: 1985 Type: Text Description: 82 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1985. URI: http://hdl.handle.net/2142/71239 Other Identifier(s): (UMI)AAI8600315 Date Available in IDEALS: 2014-12-16 Date Deposited: 1985
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