Wiener-Hopf factorisation and Fredholmness of Toeplitz operators on abstract Hardy spaces

Let $X$ be a Banach function space over the unit circle $\mathbb{T}$, let <br/>$X'$ be its associate space, and let $H[X]$ and $H[X']$ be abstract Hardy <br/>spaces built upon $X$ and $X'$, respectively. Suppose the Riesz projection<br/>$P$ is bounded from $X$ onto $H[X]$. We say that a function $a$ invertible <br/>in $L^\infty$ admits a Wiener-Hopf factorisation in $X$ if it can be written <br/>as $a=a_-\mathbf{e}_\kappa a_+$, where the middle factor is<br/>$\mathbf{e}_\kappa(t)=t^\kappa$ with $t\in\mathbb{T}$ and <br/>$\kappa\in\mathbb{Z}$, the outmost factors satisfy<br/>$$<br/>\overline{a_-}\in H[X],<br/>\quad <br/>\overline{a_-^{-1}}\in H[X'], <br/>\quad<br/>a_+\in H[X'],<br/>\quad<br/>a_+^{-1}\in H[X],<br/>$$ <br/>and the operator $P$ is bounded on the weighted space $X(|a_+^{-1}|)$.<br/>We prove that a Toeplitz operator $T(a)f:=P(af)$ with symbol $a\in L^\infty$<br/>is Fredholm on $H[X]$ if and only if $a$ is invertible in $L^\infty$ and admits <br/>a Wiener-Hopf factorisation in $X$. In this case, the Fredholm index of <br/>$T(a)$ is equal to $-\kappa$. No assumptions on separability or reflexivity <br/>of $X$ are made, which significantly complicates the matters. <br/>Our result extends a scalar version of Simonenko's factorisation theorem <br/>obtained in the 1960s for Lebesgue spaces.