{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71208"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71208","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Analytic Unitary Operators","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).","abstract_html":"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).","abstract_has_math":false,"creators":["Wingler, Eric Jeffrey"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T06:18:04Z","date_published":"2014-12-16T06:18:04Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8303024"],"render_values":[{"text":"(UMI)AAI8303024","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71208","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Wingler, Eric Jeffrey"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:04Z","10000-01-01","1982"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71208","(UMI)AAI8303024"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.","Made available in DSpace on 2014-12-16T06:18:04Z (GMT). No. of bitstreams: 1 8303024.pdf: 1538673 bytes, checksum: bcec780a4d98fe898b185c77726764af (MD5) Previous issue date: 1982","Embargo set by: Seth Robbins for item 71374 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","65 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1982."]},{"key":"dc:title","label":"Title","values":["Analytic Unitary Operators"]}]}],"canonical_facts":{"dc:creator":["Wingler, Eric Jeffrey"],"dc:date":["2014-12-16T06:18:04Z","10000-01-01","1982"],"dc:description":["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.","Made available in DSpace on 2014-12-16T06:18:04Z (GMT). No. of bitstreams: 1 8303024.pdf: 1538673 bytes, checksum: bcec780a4d98fe898b185c77726764af (MD5) Previous issue date: 1982","Embargo set by: Seth Robbins for item 71374 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","65 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1982."],"dc:identifier":["http://hdl.handle.net/2142/71208","(UMI)AAI8303024"],"dc:subject":["Mathematics"],"dc:title":["Analytic Unitary Operators"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:04Z"}