Exercise 21, § 1

Let $D$ denote the open disc with center 0 and radius 1 in $\mathbf{C}$ and let $H^2(D)$ denote the Hardy space on D (EVT, V, p. 7, example 5). We write $\mathbf{T}=\mathbf{R}/2\pi \mathbf{Z}$, which we endow with the normalized Haar measure. Let $j$ be the isometry of $H^2(D)$ into $L^2(\mathbf{T})$ defined by

$j(f)(\theta ) =\sum f^(\overset{n)}{n}$!$(0)e^{in\theta}$.

$n\geqslant 0$

The image of $H^2(D)$ by $j$ will be denoted by $H^2(\mathbf{T})$.

Let $\widetilde{P}$ be the orthoprojector of $L^2(\mathbf{T})$ with image $H^2(\mathbf{T})$ and let P be the linear mapping $f\mapsto \widetilde{P}(f)$ of $L^2(\mathbf{T})$ into $H^2(\mathbf{T})$. If $f$ is an element of $L^{\infty}(\mathbf{T})$, we denote by $M_f$ the endomorphism $g\mapsto f g$ of $L^2(\mathbf{T})$.

Prove that if $f$ is a continuous function, then the commutator $\widetilde{P}M_f-$ $M_f\widetilde{P}$ is a compact linear mapping. (Consider the particular case where $f$ is a trigonometric polynomial.)