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

ISBN-10: 3662026546

ISBN-13: 9783662026540

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Put q; = a/lal. 19, T(a) is invertible if and only if T(q;) is so. Let T(q;) be invertible. 22 there is an h E GHoo such that ph is sectorial. Thus, ph = Ihl e- iD with some real-valued function v E L'lO for which JlvJloo < 1':12. Hence q; = (Ihl/h) eiD = (hllhl) eiD. 40 (g), (ii)). The real part of log h is u(z):= log Ih(z)1 (z E D) and log h can be written as log h(z) = u(z) + ill(z) + ie (z E D) with some e E 1R.. Consequently, h(z) = eU(Z1 ei("(%l+c) (z E D). e. on T and equals Ih(t)I. e. on T and is equal to h(t)/lh(t)I.

11 (g)). 11 for p = 2, is the same as 1 > dist (ip, H"") = inf {Jlq5 - kll",,: h EH""} = inf {1lq5 - kll",,: hE H""} (b) Since T*(tp) = = inf {JItp - hll",,: h EH""} = dist (tp, H""). T(ip), this is immediate from (a). (c) Suppose T(tp) E Gl'(H2). Then, by (a), there is an hE Haa such that Iltp - hll"" and it remains to show that h E GH"". 8(a» Jlf - T(iph)1l = Jl1 - iphll"" = Jltp - hll"" < < 1 (1) 1, this implies the invertibility of T(iph) = T(ip) T(h) = (2) T*(tp) T(h) , and because T*(tp) is invertible, so also is T(h).

R 2n let I/(C) -Irl dm(C), Ma(f):= sup III~" /11 ~ I $; where 1 ranges over subarcs of T only, and put Mo(f):= lim Ma(/) , ~ A function I E LI is said to have bounded mean oseillation, or to be in BMO, if lI/lI. < 00, and is said to have vaniBking mean oseillation, or to belong to VMO, if lI/li. < 00 and Mo(f) =0. 47. Basic properties 01 BUO snd VMO. (a) H I E LI snd if for each subaro 10fT with 111 that (1/1/1) 11- lXII dm ~ M, then Ma(f) ::s:; 2M. J ~ ~ there is a constant lXI such I (b) H / E LI and Ma(f) < 00, where 0 < X M,,(f) (the estimate is very generous).

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