Spectral theory of Toeplitz and Wiener-Hopf operators

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There are at least two reasons for the continuous and increasing interest in the Riemann-Hilbert problem, Toeplitz and Wiener-Hopf operators. The first one is that the methods of the Riemann-Hilbert problem represent a universal and a most powerful tool that has been applied to a wide variety of problems in elasticity theory, fluid dynamics, physics, geometry, combinatorics, integrable systems, orthogonal polynomials, random matrices, probability and stochastic processes, information and control theory, and in many other fields. On the other hand, Toeplitz and Wiener-Hopf operators constitute one of the most important classes of non-self-adjoint operators with a very rich spectral theory, which utilizes methods of operator theory, function theory and the theory of Banach algebras.We intend to achieve a breakthrough in the study of a very wide class of Toeplitz operators. We also intend to make a major contribution to the study of asymptotic behaviour for truncated Wiener-Hopf operators, which is a field attracting a great deal of interest at present.