Generalized inverses and solution of equations with Toeplitz plus Hankel operators. (English) Zbl 1394.47034
For \(1<p<\infty\), let \(H^p(\mathbb{T})\) denote the Hardy space on the unit circle \(\mathbb{T}\). Assuming that \(a, b\in L^\infty(\mathbb{T})\) and \(a(t)a(1/t) = b(t) b(1/t)\), \(t\in \mathbb{T}\), the authors develop two analytic methods to solve the following Toeplitz plus Hankel equation:
\[
(T_a +H_b)\varphi = f, \quad f\in H^p(\mathbb{T}).
\]
Both constructions use generalized inverses and Wiener-Hopf factorizations. The first one is related to a Toeplitz operator \(T_{V(a, b)}\), where \(V(a, b)\) is a triangular matrix-valued function; the second one uses generalized inverses of the initial operator \(T_a +H_b\).
Reviewer: Evgueni Doubtsov (St. Petersburg)
MSC:
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
| 47A50 | Equations and inequalities involving linear operators, with vector unknowns |
Keywords:
Toeplitz plus Hankel operators; generalized inverses; closed form solutions; non-homogeneous equationReferences:
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