Analysis of Toeplitz Operators by Dr. rer. nat. Albrecht Böttcher, Prof. Dr. sc. nat. Bernd

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

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

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R(IP(Zl). Proof. It remains to show that MP is complete. Let {a(ml}mez+ be a Cauchy sequence in MP. By virtue of (d), {a(ml} is a Cauchy sequence in L'Xl and, consequently, there is an a E Loo such that Ha(ml - all oo -+ 0 as m -+ 00. Since l'(lP(Z») is complete, there exists an A E l'(lP(Z») such that JlM(a(ml) - AJlp -+ 0 as m -+ 00. Because M(a) P = Ap for P E lO(Z), we finally conclude that A = M(a). 6. Definition. r(HPl ~ cp JlaJloo, where cp is the norm of the Riesz projection on LP. This operator is called the Toeplitz operator on HP generated by the 51 Toeplitz operators function a.

Definition. Let a E L oo . e. on T, then ais called sectorial. It is obvious that a sectorial function is necessarily invertible in Loo. It is also easy to see that 58 2. Basic theory ais seetorial if and only if ajlal is so. Thus, seetoriality is a matter of the argument. 7+C), where e E 1R and v E LClO is a real-valued function with JlvJloc < 7tj2. Finally, it is easily verified that a function a E GLClO is sectorial if and only if dist LClO (aJlal, CC) < 1, where dist v '" (g, CC) := inf {J/g - eUClO: e E CC}.

It is obvious that hE GHoo and that q; = (l/h) tph, whence T(q;) = T(l/h) T(tp) T(h). 17. Thus, T(q;) is invertible, too. 24. Remark. The Widom-Devinatz theorems solve the invertibility problem in H2 for Toeplitz operators (with symbols in GLOO) completely. 23(1). This is the reason for a great part of all further investigations devoted to the invertibility of Toeplitz operators. The main goal of these investigations is to obtain invertibility criteria, or, equivalently, descriptions of the spectrum, in terms of geometrie data of the symbol.

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