Proposition 24

Let $g$ be a $\mu$-measurable function on X.

a) One has $m_{\overline{g}}\circ m_g=m_{|g|^2}$;

b) For all integers $k, \ell \in \mathbf{N}$, one has $m_{g^k\overline{g}^{\ell}}=m^k_gm^{\ell}_{\overline{g}}$.

One has $m_{\overline{g}}\circ m_g\subset m_{|g|^2}$ by Lemma 6. Conversely, one deduces from the inequality $|g|\leqslant 1 +|g|^2$ that dom($m_{|g|^2}$) $=$ dom($m_{\overline{g}}\circ m_g$), whence the first assertion.

Let $k, \ell \in \mathbf{N}$. We have $m^k_gm^{\ell}_{\overline{g}}\subset m_{g^k\overline{g}^{\ell}}($loc. cit.). The domain of $m^k_gm^{\ell}_{\overline{g}}$ is the set of classes in $L^2(X, \mu)$ of functions $h\in \mathscr{L}^2(X, \mu)$ such that $|g|^jh$ belong to $\mathscr{L}^2(X, \mu)$ for every integer $j$ such that $0\leqslant j\leqslant k+\ell$. The inequalities

valid for $0\leqslant j\leqslant k+\ell$, show that dom($m^k_gm^{\ell}_{\overline{g}}$) is equal to dom($m_{g^k\overline{g}^{\ell}}$), whence assertion b).