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.
Original language | English |
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Award date | 4 Nov 2013 |
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