Toeplitz plus Hankel operators T (a)+H(b), a, b ∈ L ∞ acting on the classical Hardy spaces H p , 1 < p < ∞, are studied. If the generating functions a and b satisfy the so-called matching conditionan effective description of the structure of the kernel and cokernel of the corresponding operator is given. The results depend on the behaviour of two auxiliary scalar Toeplitz operators, and if the generating functions a and b are piecewise continuous, more detailed results are obtained.

Abstract. The stability of the Nyström method for the Sherman-Lauricella equation on piecewise smooth closed simple contour Γ is studied. It is shown that in the space L 2 the method is stable if and only if certain operators associated with the corner points of Γ are invertible. If Γ does not have corner points, the method is always stable. Numerical experiments show the transformation of solutions when the unit circle is continuously transformed into the unit square, and then into various rhombuses. Examples also show an excellent convergence of the method.

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Wiener-Hopf plus Hankel operators W(a)+ H(b) : Lp(R+) ? Lp(R+) with generating functions a and b from a subalgebra of L?(R) containing almost periodic functions and Fourier images of L1(R)-functions are studied. For a and b satisfying the so-called matching condition a(t)a(?t) = b(t)b(?t), t ? R, we single out some classes of operators W(a)+ H(b) which are subject to the Coburn-Simonenko theorem.