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