Asymptotics of Toeplitz Matrices with Symbols in Some Generalized Krein Algebras

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Let alpha,beta is an element of (0,1) and K-alpha,K-beta := \{a is an element of L-infinity(T): Sigma(infinity)(k=1) vertical bar(a) over cap(-k)vertical bar(2)k(2 alpha) < infinity, Sigma(infinity)(k=1) vertical bar(a) over cap (k)vertical bar(2)k(2 beta) < infinity\}. Mark Krein proved in 1966 that K-1/2,K-1/2 forms a Banach algebra. He also observed that this algebra is important in the asymptotic theory of finite Toeplitz matrices. Ten years later, Harold Widom extended earlier results of Gabor Szego for scalar symbols and established the asymptotic trace formula trace f(T-n(a)) = (n+1)G(f)(a) + E-f(a) + o(1) as n ->infinity for finite Toeplitz matrices T-n(a) with matrix symbols a is an element of K-NXN(1/2,1/2). We show that if alpha + beta >= 1 and a is an element of K-NXN(alpha,beta), then the Szego-Widom asymptotic trace formula holds with o(1) replaced by o(n(1-alpha-beta)).
Original languageUnknown
Title of host publicationOperator Theory Advances and Applications
EditorsV Adamyan, Y Berezansky, I Gohberg, M Gorbachuk, V Gorbachuk, A Kochubei, H Langer, G Popov
Place of PublicationVIADUKSTRASSE 40-44, PO BOX 133, CH-4010 BASEL, SWITZERLAND
ISBN (Print)978-3-7643-9918-4
Publication statusPublished - 1 Jan 2009
EventInternational Conference on Modern Analysis and Applications -
Duration: 1 Jan 2007 → …


ConferenceInternational Conference on Modern Analysis and Applications
Period1/01/07 → …

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