Hardy space of the circle ($H^2(\mathbb{T})=\overline{\mathrm{span}}\{z^j:j\ge 0\}$)

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 $f,g$, $T_f{T}_g-T_{fg}$ is compact: it is finite rank for Laurent polynomials, and uniform approximation extends the result.