By Albrecht Böttcher

Award-winning monograph of the Ferran Sunyer i Balaguer Prize 1997.

This e-book is a self-contained exposition of the spectral conception of Toeplitz operators with piecewise non-stop symbols and singular imperative operators with piecewise non-stop coefficients. It comprises an advent to Carleson curves, Muckenhoupt weights, weighted norm inequalities, neighborhood ideas, Wiener-Hopf factorization, and Banach algebras generated via idempotents. a few simple phenomena within the box and the ideas for treating them got here to be understood purely in recent times and are comprehensively offered right here for the 1st time.

The fabric has been polished with the intention to make complex subject matters available to a vast readership. The ebook is addressed to a large viewers of scholars and mathematicians drawn to genuine and intricate research, sensible research and operator theory.

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**Extra resources for Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators**

**Sample text**

Finally, put fm,i:=f(ti,Em(ti )), in case Am -=I- 0, and let Em rm,i:=f(ti ,5Em(ti)), = 0 in case Am = Em:=Urm,i 0. 24. 5). 40) for all t E Q and all E E [EO, 11co]. Proof. 7. 41) always holds in case f is unbounded. If t E Q and E 2:: co, then Q is a subset of the portion f(t, 2E). 20 imply that If(t, E)I", If(t,2E)1 If(t,2E)1v 0 If(t,2E)I", < . < 2 r ,"=,";----:'-:Ir(t, E)I - If(t, E)I If(t,2E)1 If(t,2E)1 ",=,-:-,-:"";-'- 2 Or If(t,2E)I", . If(to, Eo)1 . If(to, Eo)l", If(to, Eo)l", lr(t,2E)1 If(to, Eo)1 < 20r cP( If(t, 2E)1 w If(to,Eo)1 )P If(to, Eo)1 If(t,2E)1 .

9. Let ~o = [0,1] and let Wo E Ap(~o) be a power weight. Extend Wo by symmetric reproduction with lk = 1/2 k to a weight vo on the segment Thus, if wo has both a pole and a zero, then vo is a weight in Ap([O, 2]) such that limsupvo(x) = +00, liminfvo(x) x~2 x~2 = 0. Now take vo E Ap([O, 2]) as the mother weight and extend it by symmetric reproduction with lk = 1/2 k to a Muckenhoupt weight uo on the segment The strange singularity of vo at the point x = 2 is then repeated a countable number of times by uo.

Throughout what follows, we define q E (1,00) by l/p + l/q = 1. We denote by Ap(r) the set of all weights w : f ---+ [0,00] such that 27 A. , Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators © Birkhäuser Verlag 1997 Chapter 2. e f W(T)P1dTI)1/P(! 1) r(t,E) where, as in Chapter 1, f(t,e) := {T E f: IT-tl < e}. 1) is referred to as the Muckenhoupt condition and weights in Ul