Titlebook: Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators; Albrecht Böttcher,Yuri I. Karlovich Book 1997 Springer Basel AG 1997 Singula

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General properties of Toeplitz operators,er words, .(a) is the bounded operator which sends . ∈ ..(Γ, ω) to .(ag) ∈ ..(Γ, ω). A central problem in the spectral theory of singular integral operators is the determination of the essential spectrum of Toeplitz operators with piecewise continuous symbols. This problem will be completely solved in Chapter 7.

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Carleson curves,xamples. The “oscillation” of a Carleson curve Γ at a point . ∈ Γ may be measured by its Seifullayev bounds ..and ..as well as its spirality indices .. and ..The definition of the spirality indices requires the notion of the W transform and some facts from the theory of submultiplicative functions.

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Boundedness of the Cauchy singular integral,of this book, says that . is bounded on ..(Γ, .) (1 <. ∞) if and only if Γ is a Carleson curve and . is a Muckenhoupt weight in ..(Γ). The proof of this theorem is difficult and goes beyond the scope of this book. We nevertheless decided to write down a proof, but this proof will only be given in Ch

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Piecewise continuous symbols,bols of the local representatives, we will completely identify the essential spectra of Toeplitz operators with piecewise continuous symbols. We know from the preceding chapter that the essential spectrum of a classical Toeplitz operator is the union of the essential range of the symbol and of line

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