University of Illinois Urbana-Champaign

Analytic Unitary Operators

Wingler, Eric Jeffrey

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Description

Title
Analytic Unitary Operators
Author(s)
Wingler, Eric Jeffrey
Issue Date
1982
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
  • 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 one-parameter 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 non-separable 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.
Graduation Semester
1982
Type of Resource
text
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http://hdl.handle.net/2142/71208

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