Corollary

Let $g\in L^{\infty}(X, \mu)$.

a) The spectrum of $\widetilde{m}_g$ in $\mathscr{L}(L^2(X, \mu))$ is the μ-essential image of $g$;

b) The endomorphism $\widetilde{m}_g$ is positive if and only if $g$ is locally μ-almost everywhere with positive values;

c) For every function $f\in \mathscr{C}$ (Sp($\widetilde{m}_g$)), one has $f(\widetilde{m}_g) =\widetilde{m}_{f\circ g}$.

Since $\widetilde{m}$ is injective, one has Sp($\widetilde{m}_g$) $=$ Sp($g$) by Prop. 5 of I, p. 106; the result then follows from Lemma 6 of IV, p. 185, from Proposition 4 of IV, p. 185 and from Definition 6 of I, p. 115.