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Title:  Analytic Unitary Operators 
Author(s):  Wingler, Eric Jeffrey 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  Forelli has shown that every linear isometry T from H('p) onto H('p),1 (LESSTHEQ) p < (INFIN), p (NOT=) 2, is of the form Tf = (nu)((phi)')('1/p)f(CCIRC)(phi), where (nu) is a unimodular constant and (phi) is in M, the group of Mobius transformations of the unit disc . For p = 2, the operators of this form are called analytic unitary operators. These operators form a group, which is denoted by OU(,S) and is a proper subgroup of the group U of unitary operators on H('2). The main objective of this work is to investigate operators in OU(,S). The analytic unitary operators are distinguished from the other operators in U by their relation to the shift operator S defined by (Sf) (z) = zf(z) for f in H('2). In Fact, T is in OU(,S) if and only if there is an element (phi) in M such that TST* = (phi)(S). Besides this property, a unitary operator T can be characterized as an analytic unitary operator by either of the following: (1) T can be expressed as the composition of a multiplication operator and a multiplicative operator; (2) TST* commutes with S. A means is given by which the spectra of elements of OU(,S) can be computed and also given is the spectral decomposition of oneparameter groups of analytic unitary operators. In the uniform operator topology, OU(,S) is nowhere dense in U andalso nonseparable. Although OU(,S) is not a normal subgroup of U, thequotient topological space U/OU(,S) = {{U} : U (ELEM) U}, where {U} ={UT : T (ELEM) OU(,S)}, can still be considered. If U has the uniform operator topology, then U/OU(,S) is nonseparable in the quotient topology. Operators of the form Tf = ((PHI)')(' 1/2)f(CCIRC)(PHI), where (PHI) is a Mobius transformation mapping into , are also considered. The normal operators of this form are characterized. 
Issue Date:  1982 
Type:  Text 
Description:  65 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1982. 
URI:  http://hdl.handle.net/2142/71208 
Other Identifier(s):  (UMI)AAI8303024 
Date Available in IDEALS:  20141216 
Date Deposited:  1982 
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Dissertations and Theses  Mathematics

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