How to solve an equation with a Toeplitz operator? (Q2010507)

This is an expository paper surveying methods to solve Toeplitz operator equations \(T(a)f=g\) in \(H^2(\mathbb{T})\), the Hardy spaces of square integrable functions on the unit circle, or in \(A^2(\mathbb{D})\), the Bergman space of functions square integrable over the unit disc. In the case of \(H^2(\mathbb{T})\), one may use Wiener-Hopf factorization of the symbol \(a\) and invert the factors or one may choose an orthogonal basis \(\{e_k\}_{k=0}^\infty\) and solve the corresponding matrix representation via finite sections (i.e., project on \(\operatorname{span}\{e_k\}_{k=0}^n\) for successive \(n\)). These methods do not work on \(A^2(\mathbb{D})\) but different variants of the Galerkin-Petrov method can be applied. Therefore choose a finite basis \(\{u_k^{(n)}\}_{k=0}^n\) and test functions \(\{v_k^{(n)}\}_{k=0}^n\) and solve a system obtained via inner products \((Tf^{(n)},v_k^{(n)})=(g,v_k^{(n)})\), \(k=0,\dots,n\). These functions can be monomial powers, polynomials, or Bergman (reproducing) kernels for a set of collocation (interpolation) points. Convergence results are recalled from the literature without proof and numerical examples for \(A^2(\mathbb{D})\) illustrate the methods with special attention for condition numbers of the matrices involved. The paper starts with some personal notes about Rinus Kaashoek, as it is a chapter in a volume dedicated to his 80th birthday. For the entire collection see [Zbl 1411.47002].