Toeplitz operators whose symbols have discontinuities of the infinite index type (Q1360746)

Let \(L_p^{(n)}\) and \(L_p^{(n\times n)}\) be the \(L^p\)-spaces of vector-valued functions \(f:\mathbb{T}\to\mathbb{C}^n\) and \((n\times n)\)-matrix valued functions \(f:\mathbb{T} \to\mathbb{C}^{n\times n}\), respectively, on the unit circle \(\mathbb{T}\) with respect to the arc-length measure, \(H_p^{(n)} \subset L_p^{(n)}\) the corresponding Hardy spaces and \(P_+\) the Riesz projection. The paper gives several results concerning the (left- and right-semi-) Fredholmness of the Toeplitz operators \(T_A: f\mapsto P_+(Af)\) on \(H_p^{(n)}\), \(1<p<\infty\), for functions \(A\in L_\infty^{(n\times n)}\) possessing certain factorization properties; in particular, for a class of functions of the form \(A(t)=\{c_{ij}(t) g_{ij}(\exp(2\pi i f(t)))\}_{i,j=1}^n\), where \(c_{ij}\) and \(g_{ij}\) are continuous functions on \(\mathbb{T}\) while \(f\) is continuous on \(\mathbb{T}\setminus\{1\}\) and satisfies \(\lim_{ \theta\to\pm 1} f(e^{i\theta}) = \mp\infty\). The precise hypotheses on \(f\) are too complicated to be reproduced here, but examples include \(f(e^{i\theta})=\ln\theta\), or \(f(e^{i\theta})= -\theta^{-\lambda}| \ln\theta| ^\beta\), \(\lambda>0,\beta\in\mathbb{R}\), or \(f(e^{i\theta})=-\exp(\exp(\theta^{-\lambda}))\), \(\lambda>0\) (in all cases, the equality is assumed to hold for all sufficiently small \(\theta>0\), and \(f(e^{-i\theta})=-f(e^{i\theta})\)). Proofs are not given.