Exercise 7, § 5

One equips the unit circle $\mathbf{S}_1\subset \mathbf{C}$ with the normalized Haar measure. Let E be the complex Hilbert space $L^2(\mathbf{S}_1)$. For $n\in \mathbf{Z}$, let $e_n$ be the element $z\mapsto z^n$ of E. Let $F\subset E$ be the closed subspace generated by the $e_n$ for $n\geqslant 0$, and let $p: E\rightarrow F$ be the orthogonal projector. For $f\in L^{\infty}(\mathbf{S}_1)$, one defines linear mappings $M_f$ (resp. $T_f$) of E (resp. F) by $M_f(g) =f g$ for $g\in E$ and $T_f=p\circ M_f$. One says that $T_f$ is the Toeplitz operator with symbol $f$.

a) The linear mappings $M_f$ and $T_f$ are continuous. The spectrum of $M_f$ is the closure of the set of Fourier coefficients $c_n=\langle f|e_n\rangle$ of $f$. The endomorphism $M_f$ is a Fredholm endomorphism if and only if it is a Riesz endomorphism.

b) The endomorphisms $T_{e_n}$ for $n\in \mathbf{Z}$ are Fredholm endomorphisms and ind(T$_{e_n}$) $=-n$ for all $n\in \mathbf{Z}$.

c) One has $T_{\overline{f}}= T^*_f$ for every mapping $f\in L^{\infty}(\mathbf{S}_1)$.

d) If $T_f$ is an isomorphism, then $M_f$ is an isomorphism and $f$ is invertible in $L^{\infty}(\mathbf{S}_1)$. Deduce that the spectrum of $M_f$ is included in the spectrum of $T_f$ and that $\|T_f\|=\|f\|$.

e) Let $f\in \mathscr{C}(\mathbf{S}_1)$ be a continuous mapping and let $g\in L^{\infty}(\mathbf{S}_1)$. The endomorphisms $T_fT_g-T_{f g}$ and $T_gT_f-T_{gf}$ are compact. (First consider the case where $f=e_n.$)

f ) Let $\mathscr{C}$alk(F) be the Calkin algebra of F and $\pi :\mathscr{L}(F)\rightarrow \mathscr{C}$alk(F) the canonical projection. Prove that $f\mapsto \pi (T_f)$ is an injective morphism of involutive algebras of $\mathscr{C}(\mathbf{S}_1)$ into $\mathscr{C}$alk(F). (To prove injectivity, one may use Prop. 1 of I, p. 31.)

g) Let $f\in \mathscr{C}(\mathbf{S}_1)$. Then $T_f$ is a Fredholm endomorphism if and only if $f$ does not vanish on $\mathbf{S}_1$. (Use Prop. 5 of I, p. 106.)

h) Let $f\in \mathscr{C}(\mathbf{S}_1)$ not vanishing on $\mathbf{S}_1$. The index of $T_f$ is equal to $n\in \mathbf{Z}$ if and only if the path $t\mapsto f(e^{2i\pi t})$ in $\mathbf{C}-\{0\}$ is homotopic to the path $t\mapsto e_n(t)$ (cf. TA, III, p. 229, def. 1).