Exercise 22, § 1

We retain the notation of the preceding exercise. Let R be the continuous linear mapping from $H^2(\mathbf{T})$ into $L^2(\mathbf{T})$ defined by $Rf(\theta ) =f(-\theta )$. For $g\in L^{\infty}(\mathbf{T})$, put $\Gamma_g=$ PM$_gR$. a) Prove that the linear mapping $\Gamma :g\mapsto \Gamma_g$ from $L^{\infty}(\mathbf{T})$ into $\mathscr{L}(H^2(\mathbf{T}))$ is continuous, of norm $\leqslant 1$, and that its kernel F is equal to $H^2(\mathbf{T})^{\circ}\cap L^{\infty}(\mathbf{T})$. b) If $g$ is a continuous function, then $\Gamma_g$ is a compact endomorphism of $H^2(\mathbf{T})$.

c) Let $g\in L^{\infty}(\mathbf{T})$. Denote by $\alpha_g$ the linear form on $H^2(\mathbf{T})$ defined by

Let $\mathscr{A}$ be the image under $j$ of the subspace of $H^2(D)$ consisting of the holomorphic functions which admit an analytic continuation to a neighbourhood of D. Prove that if $f_1$ and $f_2$ belong to $\mathscr{A}$, then

d) Let $\varphi$ be a holomorphic function in a neighbourhood of $\overline{D}$ which does not vanish on $\partial D$, and let $f=j(\varphi |D)$. Then there exist $f_1$ and $f_2$ in $\mathscr{A}$ such that $f=f_1f_2$ and

(Let $z_1, . . . , z_n$ be the zeros of $\varphi$ in D; then there exists a holomorphic function $\psi$ in a neighbourhood of D such that

$k=1$

for every $z.)$

e) Prove that, for every $f\in H^2(\mathbf{T})$, one has the inequality

(If $\varphi \in H^2$(D), there exists a sequence $(r_n)_{n\in\mathbf{N}}$ de$]0,1[$ such that, for every $n\in \mathbf{N}$, the function $\varphi_n$ defined by $\varphi_n(z) =\varphi (r_nz)$ has no zeros on $\partial D$, and lim$\varphi_n=\varphi$ in $H^2(D).$)

Deduce that there exists $h\in L^{\infty}(\mathbf{T})$ such that

for every $f\in H^2(\mathbf{T})$, and $\|h\|_{L^{\infty}(\mathbf{T})}\leqslant \|\Gamma_g\|_{\mathscr{L}(H^2(\mathbf{T}))}$.

f) Prove that there exists $k\in L^{\infty}(\mathbf{T})$ such that $\Gamma_g= \Gamma_k$ and $\|k\|_{L^{\infty}(\mathbf{T})}=$ $\|\Gamma_g\|_{\mathscr{L}(H^2(\mathbf{T}))}$.

The mapping $\widehat{\Gamma} : L^{\infty}(\mathbf{T})/F\rightarrow \mathscr{L}(H^2(\mathbf{T}))$ is therefore an isometry ("Nehari Theorem").

g) Let $g$ be a continuous function on $\mathbf{T}$. Then one has $d(g,F) =d(g,F\cap \mathscr{C}(\mathbf{T}))$. (For $R>1$, define the linear mapping $P_R$ of $L^{\infty}(\mathbf{T})$ into $\mathscr{C}(\mathbf{T})$ by

Then $\|P_Rf\|_{\mathscr{C}(\mathbf{T})}\leqslant \|f\|_{L^{\infty}(\mathbf{T})}$ and, if $f\in \mathscr{C}(\mathbf{T})$, one has $P_Rf\rightarrow f$ in $\mathscr{C}(\mathbf{T})$ when $R\rightarrow 1.)$

Deduce that $\Gamma (\mathscr{C}(\mathbf{T}))$ is a closed subspace of $\mathscr{L}(H^2(\mathbf{T}))$ ("Sarason's Theorem").

h) Let $g\in L^{\infty}(\mathbf{T})$ be such that $\Gamma_g$ is a compact endomorphism. For $m\in \mathbf{Z}$, put $e_m(\theta ) =e^{im\theta}$. Prove that $\|\Gamma_{ge_{-m}}\|_{\mathscr{L}(H^2(\mathbf{T}))}$ tends to zero as $m$ tends to $+\infty$, then that there exists a sequence $(h_m)_{m\in\mathbf{N}}$ of F such that the sequence $(e_mh_m)_{m\in\mathbf{N}}$ converges to $g$ in $L^{\infty}(\mathbf{T})$. Deduce that there exists a continuous function in the class of $g$ modulo F ("Hartman's Theorem").