Fredholmness and invertibility of Toeplitz operators with matrix almost periodic symbols (Q2781293)

It was proved by \textit{L. A. Coburn, R. D. Moyer} and \textit{I. M. Singer} that Toeplitz operators on \(L^2(\mathbb{R}^n)\) with (suitably interpreted) almost periodic symbols are Fredholm if and only if they are invertible [Acta Math. 130, 279-307 (1973; Zbl 0263.47042)]. In the present paper, the authors prove a generalization of this result to almost periodic matrix symbols on \(\mathbb{R}^k\). Namely, let \((AP^k)\) be the closed subalgebra in \(L^\infty(\mathbb{R}^k)\) generated by the functions \(e_\lambda(t)=e^{i\langle \lambda,t\rangle}\), \(\lambda\in\mathbb{R}^k\). For \(f\in(AP^k)\), denote \(f_\lambda=\lim_{T\to\infty} (2T)^{-k} \int_{[-T,T]^k} f(t) e_{-\lambda} (t) dt\), and for a subset \(\Omega\subset\mathbb{R}^k\) let \((AP^k)_\Omega = \{f\in(AP^k): f_\lambda=0\text{ if }\lambda\notin\Omega\}\). Finally, let \((B^k)\) and \((B^k)_\Omega\) be the completions of \((AP^k)\) and \((AP^k)_\Omega\), respectively, with respect to the inner product \(\langle f,g\rangle =(f\overline g)_0\), \(\Pi_\Omega\) the orthogonal projection of \((B^k)\) onto \((B^k)_\Omega\), and \((AP^k)_\Omega^{n\times m}\), \((B^k)_\Omega^{n\times m}\) the analogous spaces of \(n\times m\) matrices. For \(f\in(AP^k)^{n\times n}\), the corresponding Toeplitz operator \(T_f\) is defined on \((B^k)_\Omega^{n\times 1}\) by \(T_f g =\Pi_\Omega(fg)\) \(\forall g\in (AP^k)_\Omega^{n\times 1}\). The main result then is that for the set \(\Omega\) satisfying a certain condition, the Toeplitz operator \(T_f\) is Fredholm if and only if it is invertible. In particular, this holds if \(\Omega\) is an ``exact half-space'' in \(\mathbb{R}^k\), i.e. a cone closed under addition and such that \(\Omega\cup(-\Omega)=\mathbb{R}^k\) and \(\Omega \cap(-\Omega)=\{0\}\).