Asymptotics of Toeplitz Matrices with Symbols in Some Generalized Krein Algebras

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

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
PublisherBIRKHAUSER VERLAG AG
Pages341-359
Volume190
ISBN (Print)978-3-7643-9918-4
DOIs
Publication statusPublished - 1 Jan 2009
EventInternational Conference on Modern Analysis and Applications -
Duration: 1 Jan 2007 → …

Conference

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

Cite this

Karlovych, O. (2009). Asymptotics of Toeplitz Matrices with Symbols in Some Generalized Krein Algebras. In V. Adamyan, Y. Berezansky, I. Gohberg, M. Gorbachuk, V. Gorbachuk, A. Kochubei, H. Langer, ... G. Popov (Eds.), Operator Theory Advances and Applications (Vol. 190, pp. 341-359). VIADUKSTRASSE 40-44, PO BOX 133, CH-4010 BASEL, SWITZERLAND: BIRKHAUSER VERLAG AG. https://doi.org/10.1007/978-3-7643-9919-1_21