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Siskakis, Aristomenis Georgios 2014-12-16T06:18:15Z 2014-12-16T06:18:15Z 10000-01-01 1985 1985 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. Made available in DSpace on 2014-12-16T06:18:15Z (GMT). No. of bitstreams: 1 8600315.pdf: 2483586 bytes, checksum: 6a0cd1ea2a0ce90b95ab8251c6e12770 (MD5) Previous issue date: 1985 Embargo set by: Seth Robbins for item 71405 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs U of I Only 82 p. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1985. http://hdl.handle.net/2142/71239 (UMI)AAI8600315 Mathematics Semigroups of Composition Operators and the Cesaro Operator on H('p)(d) (Bergman Space, Infinitesimal Generator) text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.