Norms of Moore-Penrose inverses of Fredholm Toeplitz operators (Q2464690)

Let \(FL_{N\times N}^\infty\) be the set of all \(N\times N\) matrix functions \(a\in L_{N\times N}^\infty\) for which the Toeplitz operator \(T(a)\) is Fredholm on \(\ell_N^2\). The Moore-Penrose inverse of a Toeplitz operator \(T(a)\) is denoted by \(T(a)^+\). It is proved that the set of all functions \(a\in FL_{N\times N}^\infty\) such that \(\| T(a)^+\| >\| a^{-1}\| _\infty\) holds is an open and dense subset of \(FL_{N\times N}^\infty\). Further, if \(a\in FL_{N\times N}^\infty\) is continuous, then \( \lim_{r\to+\infty}\| T(t^ra)^+\| = \lim_{r\to+\infty}\| T(t^{-r}a)^+\| = \| a^{-1}\| _\infty. \) Finally, these results are illustrated by numerical experiments.