Maximal noncompactness of Wiener-Hopf operators

Let $X(\mathbb{R})$ be a separable translation-invariant Banach<br/>function space and $a$ be a Fourier multiplier on $X(\mathbb{R})$. We prove<br/>that the Wiener-Hopf operator $W(a)$ with symbol $a$ is maximally noncompact<br/>on the space $X(\mathbb{R}_+)$, that is, its Hausdorff measure of <br/>noncompactness, its essential norm and its norm are all equal. This equality<br/>for the Hausdorff measure of noncompactness of $W(a)$ <br/>is new even in the case of $X(\mathbb{R})=L^p(\mathbb{R})$ with $1\le p&lt;\infty$.<br/>