Abstract
Considered is the equation \((T(a)+H(b))\phi =f\), where T(a) and H(b), \(a,b\in L^\infty (\mathbb {T})\) are, respectively, Toeplitz and Hankel operators acting on the classical Hardy space \(H^p(\mathbb {T})\), \(1<p<\infty \). If the generating functions a and b satisfy the matching condition \(a(t) a(1/t)=b(t)b(1/t), \, t\in \mathbb {T}\), two efficient methods to solve the equation are proposed. The first method is based on generalized inverses of a Toeplitz operator T(V(a, b)) with a triangular generating matrix-function V(a, b) and the second one uses generalized inverses of the original operator \(T(a)+H(b)\). Both constructions use Wiener–Hopf factorization of the scalar functions \(c(t)=a(t)b^{-1}(t)\) and \(d(t)=a(t)b^{-1}(1/t)\).
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Notes
Note that assertion (b) of the said proposition contains a typo. The space \(X^*\) in the right-hand side there has to be replaced by X.
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To Sergei Grudsky on the occasion of his 60th birthday.
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Didenko, V.D., Silbermann, B. Generalized inverses and solution of equations with Toeplitz plus Hankel operators. Bol. Soc. Mat. Mex. 22, 645–667 (2016). https://doi.org/10.1007/s40590-016-0101-2
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DOI: https://doi.org/10.1007/s40590-016-0101-2