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Toeplitz operators, matrix Wiener-Hopf factorization and Riemann-Hilbert problems in a Riemann surface
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The equivalence between the study of several properties of Toeplitz operators with Hölder continuous symbols and the solvability of some associated Riemann-Hilbert problems is reviewed, as well as the close relationship of both with Wiener-Hopf factorization of Hölder continuous functions. The differences between the cases of scalar and matrix symbols are briefly presented. It is shown that an appropriate characterization of classes of symbols allows to associate to each class a Riemann surface ?, thus providing us with a tool that enables to reduce the problem of Wiener-Hopf factorization of matrix symbols to the study of the solvability of a related scalar Riemann-Hilbert problem on the above-mentioned Riemann surface. A new notion of ?-factorization is presented which enables us to solve these problems. |
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