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

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This factorization is unique up to a multiplicative constant. A remarkable property of HI functions is that log 1I1 E LI whenever I does not vanish identically. The outer function g occuring in the inner-outer factorization theorem can be obtained through (1) with 1p = 1/1. Examples of inner functions are: X,,(t) = trI (n ~ 0), 8(J(t) = exp ( at+l) -t-1 (a> 0), ba(t)

And M. Riesz theorem). A funetion in HI vanishes either almost everywhere or almost nowhere on T. In what follows we shall frequently identify functions in HP with their analytic extension in D. 40. Inner-outer factorization. e. on T. A function gEHl is said to be an outer lunction if its analytic extension can be represented in the form g(z) = c exp { - where cE T, 1p E LI, 1p says the following. 3* 1 21t ~ f 21"0 o } ei/! + z ---log1p(ei /l) df} , ei/! e. on T, log1p E LI. The inner-outer lactorization theorem 36 1.

18 we deduce that hE GH"". Now suppose h E GH"" and Jltp - hJl"" < 1. Then (1) holds and therefore the operator (2) is invertible. 18, T(h) E Gl'(H2), hence T*(tp) E G1'(H2) and thus T(tp) E Gl'(H2). 21. Lemma. Suppose B is a subset 01 L"" with the property that eb E B whenever c E ce " {O} and bEB. Let tp E L"" be a unimodular lunction. Then dist v '" (tp, B) < 1 il and only il there are a bEB and a sectoriallunction s E GL"" such that tp = bs. P roof. If distL"" (tp, B) < 1, then Jl1 - tp-1bJl"" = Jltp - bJl"" < 1 for some bEB.

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