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Fredholmness of Toeplitz operators and corona problems |
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| Authors: | MC Câmara |
| Institution: | a Departamento de Matemática, Instituto Superior Técnico, Technical University of Lisbon (TULisbon), Avenida Rovisco Pais, 1049-001 Lisbon, Portugal b Departamento de Métodos Quantitativos, Lisbon University Institute (ISCTE-IUL), Avenida das Forças Armadas, 1649-026 Lisbon, Portugal c Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA |
| Abstract: | A meromorphic analogue to the corona problem is formulated and studied and its solutions are characterized as being left-invertible in a space of meromorphic functions. The Fredholmness of Toeplitz operators with symbol G∈(L∞(R))2×2 is shown to be equivalent to that of a Toeplitz operator with scalar symbol  , provided that the Riemann-Hilbert problem  admits a solution such that the meromorphic corona problems with data  are solvable. The Fredholm properties are characterized in terms of  and the corresponding meromorphic left-inverses. Partial index estimates for the symbols and Fredholmness criteria are established for several classes of Toeplitz operators. |
| Keywords: | Riemann-Hilbert problem Wiener-Hopf factorization Corona theorem Toeplitz operator |
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