{"id":{"repo_id":"kings","oai_identifier":"oai:kclpure.kcl.ac.uk:studenttheses/1c891b23-dba5-4395-99dc-2884cbeb3bfb"},"canonical_url":"https://search.dev.ndltd.org/etd/kings/oai:kclpure.kcl.ac.uk:studenttheses/1c891b23-dba5-4395-99dc-2884cbeb3bfb","repository":{"repo_id":"kings","name":"King's College London","base_url":"https://kclpure.kcl.ac.uk/ws/oai"},"display":{"title":"Almost commuting elements of real rank zero C∗-algebras","abstract":"The purpose of this thesis is to study the following problem. Suppose that X, Y are bounded self-adjoint operators in a Hilbert space H with their commutator [X; Y ] being small. Such operators are called almost commuting. How close is the pair X; Y to a pair of commuting operators X0; Y 0? In terms of one operator A = X + iY , suppose that the self-commutator [A;A] is small. How close is A to the set of normal operators?
Our main result is a quantitative analogue of Huaxin Lin's theorem on almost
commuting matrices. We prove that for every (nn)-matrix A with kAk 6 1 there
exists a normal matrix A0 such that kA 􀀀 A0k 6 Ck[A;A]k1=3. We also establish
a general version of this result for arbitrary C-algebras of real rank zero assuming that A satises a certain index-type condition. For operators in Hilbert spaces, we obtain two-sided estimates of the distance to the set of normal operators in terms of k[A;A]k and the distance from A to the set of invertible operators.
The technique is based on Davidson's results on extensions of almost normal
operators, Alexandrov and Peller's results on operator and commutator Lipschitz
functions, and a rened version of Filonov and Safarov's results on approximate
spectral projections in C-algebras of real rank zero.
In Chapter 4 we prove an analogue of Lin's theorem for nite matrices with
respect to the normalized Hilbert{Schmidt norm. It is a renement of a previously
known result by Glebsky, and is rather elementary.
In Chapter 5 we construct a calculus of polynomials for almost commuting elements of C-algebras and study its spectral mapping properties. Chapters 4 and 5 are based on author's joint results with Nikolay Filonov.","abstract_html":"The purpose of this thesis is to study the following problem. Suppose that X, Y are bounded self-adjoint operators in a Hilbert space H with their commutator [X; Y ] being small. Such operators are called almost commuting. How close is the pair X; Y to a pair of commuting operators X0; Y 0? In terms of one operator A = X + iY , suppose that the self-commutator [A;A] is small. How close is A to the set of normal operators?<br/>Our main result is a quantitative analogue of Huaxin Lin's theorem on almost<br/>commuting matrices. We prove that for every (nn)-matrix A with kAk 6 1 there<br/>exists a normal matrix A0 such that kA &#x100000; A0k 6 Ck[A;A]k1=3. We also establish<br/>a general version of this result for arbitrary C-algebras of real rank zero assuming that A satises a certain index-type condition. For operators in Hilbert spaces, we obtain two-sided estimates of the distance to the set of normal operators in terms of k[A;A]k and the distance from A to the set of invertible operators.<br/>The technique is based on Davidson's results on extensions of almost normal<br/>operators, Alexandrov and Peller's results on operator and commutator Lipschitz<br/>functions, and a rened version of Filonov and Safarov's results on approximate<br/>spectral projections in C-algebras of real rank zero.<br/>In Chapter 4 we prove an analogue of Lin's theorem for nite matrices with<br/>respect to the normalized Hilbert{Schmidt norm. It is a renement of a previously<br/>known result by Glebsky, and is rather elementary.<br/>In Chapter 5 we construct a calculus of polynomials for almost commuting elements of C-algebras and study its spectral mapping properties. Chapters 4 and 5 are based on author's joint results with Nikolay Filonov.","abstract_has_math":false,"creators":["Kachkovskiy, Ilya"],"institution":"King's College London","degree_name":"Doctor of Philosophy","degree_level":"Doctoral Thesis","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Safarov, Yuri","Pushnitski, Alexander"],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-11-4","date_published":"2013-11-4","updated_at":"2026-07-24T02:44:40Z","subjects":[],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["oai:kclpure.kcl.ac.uk:studenttheses/1c891b23-dba5-4395-99dc-2884cbeb3bfb"],"render_values":[{"text":"oai:kclpure.kcl.ac.uk:studenttheses/1c891b23-dba5-4395-99dc-2884cbeb3bfb","href":null,"code":true}]}]},"links":{"outbound_url":"https://kclpure.kcl.ac.uk/portal/en/studentTheses/1c891b23-dba5-4395-99dc-2884cbeb3bfb","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Safarov, Yuri","Pushnitski, Alexander"]},{"key":"dc:creator","label":"Author","values":["Kachkovskiy, Ilya"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2013-11-4"]},{"key":"dc:date.issued","label":"Date","values":["2013-11-4"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["Mathematics"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["King's College London"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["https://kclpure.kcl.ac.uk/portal/en/studentTheses/1c891b23-dba5-4395-99dc-2884cbeb3bfb"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral Thesis"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["oai:kclpure.kcl.ac.uk:studenttheses/1c891b23-dba5-4395-99dc-2884cbeb3bfb","https://kclpure.kcl.ac.uk/portal/en/studentTheses/1c891b23-dba5-4395-99dc-2884cbeb3bfb"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://kclpure.kcl.ac.uk/portal/files/12780414/Studentthesis-Ilya_Kachkovskiy_2013.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The purpose of this thesis is to study the following problem. Suppose that X, Y are bounded self-adjoint operators in a Hilbert space H with their commutator [X; Y ] being small. Such operators are called almost commuting. How close is the pair X; Y to a pair of commuting operators X0; Y 0? In terms of one operator A = X + iY , suppose that the self-commutator [A;A] is small. How close is A to the set of normal operators?
Our main result is a quantitative analogue of Huaxin Lin's theorem on almost
commuting matrices. We prove that for every (nn)-matrix A with kAk 6 1 there
exists a normal matrix A0 such that kA 􀀀 A0k 6 Ck[A;A]k1=3. We also establish
a general version of this result for arbitrary C-algebras of real rank zero assuming that A satises a certain index-type condition. For operators in Hilbert spaces, we obtain two-sided estimates of the distance to the set of normal operators in terms of k[A;A]k and the distance from A to the set of invertible operators.
The technique is based on Davidson's results on extensions of almost normal
operators, Alexandrov and Peller's results on operator and commutator Lipschitz
functions, and a rened version of Filonov and Safarov's results on approximate
spectral projections in C-algebras of real rank zero.
In Chapter 4 we prove an analogue of Lin's theorem for nite matrices with
respect to the normalized Hilbert{Schmidt norm. It is a renement of a previously
known result by Glebsky, and is rather elementary.
In Chapter 5 we construct a calculus of polynomials for almost commuting elements of C-algebras and study its spectral mapping properties. Chapters 4 and 5 are based on author's joint results with Nikolay Filonov."]},{"key":"dc:title","label":"Title","values":["Almost commuting elements of real rank zero C∗-algebras"]}]}],"canonical_facts":{"dc:contributor.advisor":["Safarov, Yuri","Pushnitski, Alexander"],"dc:creator":["Kachkovskiy, Ilya"],"dc:date":["2013-11-4"],"dc:date.issued":["2013-11-4"],"dc:description.abstract":["The purpose of this thesis is to study the following problem. Suppose that X, Y are bounded self-adjoint operators in a Hilbert space H with their commutator [X; Y ] being small. Such operators are called almost commuting. How close is the pair X; Y to a pair of commuting operators X0; Y 0? In terms of one operator A = X + iY , suppose that the self-commutator [A;A] is small. How close is A to the set of normal operators?
Our main result is a quantitative analogue of Huaxin Lin's theorem on almost
commuting matrices. We prove that for every (nn)-matrix A with kAk 6 1 there
exists a normal matrix A0 such that kA 􀀀 A0k 6 Ck[A;A]k1=3. We also establish
a general version of this result for arbitrary C-algebras of real rank zero assuming that A satises a certain index-type condition. For operators in Hilbert spaces, we obtain two-sided estimates of the distance to the set of normal operators in terms of k[A;A]k and the distance from A to the set of invertible operators.
The technique is based on Davidson's results on extensions of almost normal
operators, Alexandrov and Peller's results on operator and commutator Lipschitz
functions, and a rened version of Filonov and Safarov's results on approximate
spectral projections in C-algebras of real rank zero.
In Chapter 4 we prove an analogue of Lin's theorem for nite matrices with
respect to the normalized Hilbert{Schmidt norm. It is a renement of a previously
known result by Glebsky, and is rather elementary.
In Chapter 5 we construct a calculus of polynomials for almost commuting elements of C-algebras and study its spectral mapping properties. Chapters 4 and 5 are based on author's joint results with Nikolay Filonov."],"dc:identifier":["oai:kclpure.kcl.ac.uk:studenttheses/1c891b23-dba5-4395-99dc-2884cbeb3bfb","https://kclpure.kcl.ac.uk/portal/en/studentTheses/1c891b23-dba5-4395-99dc-2884cbeb3bfb"],"dc:identifier.uri":["https://kclpure.kcl.ac.uk/portal/files/12780414/Studentthesis-Ilya_Kachkovskiy_2013.pdf"],"dc:language":["eng"],"dc:publisher.department":["Mathematics"],"dc:publisher.institution":["King's College London"],"dc:relation.isreferencedby":["https://kclpure.kcl.ac.uk/portal/en/studentTheses/1c891b23-dba5-4395-99dc-2884cbeb3bfb"],"dc:title":["Almost commuting elements of real rank zero C∗-algebras"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral Thesis"],"dc:type.qualificationname":["Doctor of Philosophy"]},"updated_at":"2026-07-24T02:44:40Z"}