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By Ahmad Al-Salman, Hussain Al-Qassem (auth.), Stefan Samko, Amarino Lebre, António F. dos Santos (eds.)

These lawsuits contain a wide a part of the papers provided on the In­ ternational convention Factorization, Singular Operators and comparable difficulties, which was once held from January 28 to February 1, 2002, on the collage of th Madeira, Funchal, Portugal, to mark Professor Georgii Litvinchuk's 70 start­ day. specialists in numerous fields got here to this convention to pay tribute to the nice achievements of Professor Georgii Litvinchuk within the improvement of vari­ ous components of operator idea. the most topics of the convention have been focussed round the concept of singular variety operators and factorization difficulties, yet different themes akin to power conception and fractional calculus, to call yet a pair, have been additionally offered. The aim of the convention was once to collect mathematicians from var­ ious fields inside of operator conception and serve as conception which will spotlight contemporary advances in difficulties a lot of that have been initially studied via Profes­ sor Litvinchuk and his clinical institution. A moment goal was once to stimulate in­ ternational collaboration even extra and advertise the interplay of alternative methods in present study in those parts. The complaints may be of serious curiosity to researchers in Operator The­ ory, genuine and complicated research, practical and Harmonic research, strength concept, Fractional Calculus and different parts, in addition to to graduate scholars trying to find the newest results.

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Hirschman) states that if fEW n F£i/2 has no zeros on T and winding number zero, then the determinants Dn(f) := det Tn (f) have the asymptotic behavior Dn(f) = G(ft E(f)(l + 0(1)) as n -+ 00, where the constants G(f) and E(f) are given by 2:: k(log f)dlog f)-k, 00 G(f) = exp(log f)o, E(f) = exp k=l (log f)j being the jth Fourier coefficient of log f; see [6] for details. 46 A. BOTTCHER AND S. GRUDSKY We now turn to an extension of the Szego limit theorem to symbols whose winding number about the origin is -lor 1.

Parts 1, 2, Pitman Monographs and Surveys in Pure and Applied Mathematics 94, 95. Longman, Harlow, 1998. [4] Antonevich, A. , Functional Differential Equations: I. C· -theory, Pitman Monographs and Surveys in Pure and Applied Mathematics, 70. Longman Scientific & Technical, Harlow, 1994. , Nonautonomous Dynamical Systems. Shtiintsa, Kishinev, 1984 (in Russian). [6] Kravchenko, V. G. and Litvinchuk, G. , Introduction to the Theory of Singular Integral Operators with Shift, Mathematics and its Applications, 289.

Let < a < ~. 1) is bounded. For simplicity we assume that B(z) == 0. Then it is easy °° 33 ELLIPTIC SYSTEMS AND SINGULAR WEIGHT INTEGRAL OPERATORS to verify that the functions w(z) = Tcp = eTAT(Fe- TA ) and cp are solutions to the equations W z = A(z)w and cp = ATcp + F, respectively. 1) is proved. 3): IIWzllp,n= IIWzIIp,fl = IIAeTAT(Fe- TA ) + Fllp,n ~ IIAeTAT(Fe-TA)llp,n + 1lFllp,n ~ IIAI12,nlleTAT(Fe-TA)II~,n + 1lFllp,n CIIAI12,nll eTA lli,n Ile-TAlli,nllFllp+e,n +llFllp,n ~ CllFllp+e,n. ~ Here £0 = £O(a) > 0 is an arbitrary sufficiently small number and 8 = £Op/(£O + p)(p + 1).

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