= Hardy space of the circle
{c}
{title2=$H^2(\mathbb T)=\overline{\operatorname{span}}\{z^j:j\geq0\}$}
The Hardy space of the circle is the closed subspace of <L2 space> whose negative <Fourier coefficients> vanish. The <orthogonal projection> onto it is denoted $P_+$. A continuous symbol $f$ defines a <Toeplitz operator> $T_f=P_+M_f|_{H^2}$. For continuous $f,g$, $T_fT_g-T_{fg}$ is compact: it is finite rank for Laurent polynomials, and uniform approximation extends the result.
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