Read e-book online Analysis of Toeplitz Operators PDF

By Dr. rer. nat. Albrecht Böttcher, Prof. Dr. sc. nat. Bernd Silbermann (auth.)

ISBN-10: 366202652X

ISBN-13: 9783662026526

ISBN-10: 3662026546

ISBN-13: 9783662026540

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The M(a) we have just defined on IP(Z) is sometimes also called the Laurent operator generated by a. For rp E IP(Z) and'IjJ E Iq(,Z) (l/p l/q = 1) put + (rp, 'IjJ) : = E rpn1pn· neZ It ia easy to verify that (M(a) ei' ek) = ak-i for everya E MP. Here is a converse of this. 4. Proposition. Let A E l'(IP(Z») (1 ;;::; p < (0) and 8upp08e (Aei' ek) = ak-i lor 80me 8equence {an}neZ 01 complex number8. Then there i8 an a E MP 8uch that A = M(a) and {an} i8 the Fourier coefficient 8equence 01 a. Proof.

0) Loo c:: BMO and ll/JI. (d) H / E LI and JI/Jl. (e) L'''' c:: BMO c:: ~ 1111100 for ~ < 2n, then / E BMO and M 21t(/) I E Loo. = 0, then / = const. n u, C c:: VMO c:: BMO. l::;;p< 00 $; (2nj~)6 41 BMOand VMO X + (f) BMO is a Banach space under the norm 11/1IBMo:= II/JI. 1101, where 10 = (1/21t') 1dm. VMO is a closed subspace of BMO and coincides with the closure of (P in the J T BMO norm. There are absolute constants Al' A 2 > 0 such that for all a E BMO AIMo(a) ~ dist BMo (a, VMO) ~ A 2 M o(a).

14, T(aJlal) = T(k) T(a) X T(h). Due to the preceding proposition T(h) and T(k) = T*(h) are invertible on H2 and this gives the assertion at onee. • Hemark. This proposition reduees the invertibility problem for Toeplitz operators on H2 to tbe ease of unimodular symbols. In other words, the invertibility of a Toeplitz operator on H2 is exelusively dictated by the behavior of the argument of its symbol. 20. Theorem (WIDOMjDEVINATZ). e. on T. Then = 1 < 1, i8 right-invertible on H2 ~ distLoc (q;, RClO) < 1, is invertible on H2 ~ distv '" (q;, GHClO) < 1.

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Analysis of Toeplitz Operators by Dr. rer. nat. Albrecht Böttcher, Prof. Dr. sc. nat. Bernd Silbermann (auth.)

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