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 X',Y'? 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 (n x n)-matrix A with ||A|| ≤ 1 there exists a normal matrix A' such that ||A-A'|| ≤ C||[A,A*]||¹/³. We also establish a general version of this result for arbitrary C*-algebras of real rank zero assuming that A satisfies 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 ||[A,A*]|| 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 refined 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 finite 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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:628423 |
| Date | January 2013 |
| Creators | Kachkovskiy, Ilya |
| Publisher | King's College London (University of London) |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://kclpure.kcl.ac.uk/portal/en/theses/almost-commuting-elements-of-real-rank-zero-calgebras(1c891b23-dba5-4395-99dc-2884cbeb3bfb).html |
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