Proposition 6

a) Let $g'$ be a $\mu$-measurable function on X such that $|g|\leqslant |g'|$. One has dom($\widetilde{m}_{g'}$)$\subset$ dom($\widetilde{m}_g$) and dom($\widetilde{m}_{g'}$) is a core of the partial operator $\widetilde{m}_g$;

b) Let F be a subspace of $\mathscr{L}_K^2(X, \mu)$ whose intersection with $\mathscr{K}(X; K)$ is dense in $\mathscr{K}(X; K)$ and whose image G in $L^2_K(X, \mu)$ is contained in dom($\widetilde{m}_g$). If $|g|^2$ is locally $\mu$-integrable, then $\mathscr{K}(X; K)$ is contained in the domain of $\widetilde{m}_g$ and G is a core of $m_g$.

Let us prove a). If $f\in \mathscr{L}_K^2(X, \mu)$ belong to the domain of $m_{g'}$, so that $f g'\in \mathscr{L}_K^2(X, \mu)$, the assumption implies that $f g\in \mathscr{L}_K^2(X, \mu)$, whence the result.

Let us prove that dom($\widetilde{m}_{g'}$) is a core of $\widetilde{m}_g$, that is to say that the domain of $\widetilde{m}_{g'}$ is dense in the Hilbert space $E_{\widetilde{m}_g}$. Let $h\in \mathscr{L}_K^2(X, \mu)$

whose class $\widetilde{h}$ belong to $E_{\widetilde{m}_g}$ and is orthogonal to dom($\widetilde{m}_{g'}$). This means that

for every function $h'\in \mathscr{L}_K^2(X, \mu)$ whose class $\widetilde{h}'$ belongs to dom($\widetilde{m}_{g'}$).

Let C be a compact subset of X and let $\varphi$ be its characteristic function. Let $n\in \mathbf{N}$. Denote by $\varphi_n$ the characteristic function of the $\mu$-integrable set $C_n$ of the $x\in C$ such that $|h(x)|\leqslant n$ and put $h'_n=\varphi_nh$. The class of $h'_n$ belongs to the domain of $\widetilde{m}_{g'}$ since $|g'h'_n|\leqslant n\varphi$; hence

This implies that $h$ is zero for $\mu$-almost every $x\in C_n$ and therefore, since $n$ is arbitrary, that $h$ is zero for $\mu$-almost every $x\in C$. It follows finally that $h$ is zero $\mu$-almost everywhere, since C is arbitrary and $h$ is moderate (INT, V, p. 9, § 1, n$^o3$, cor.).

Let us now consider assertion b). Since $|g|^2$ is locally $\mu$-integrable, the function $f g$ belongs to $\mathscr{L}_K^2(X, \mu)$ if $f\in \mathscr{K}(X; K)$, hence $\mathscr{K}(X; K)$ is contained in the domain of $\widetilde{m}_g$.

Let $h\in \mathscr{L}_K^2(X, \mu)$ whose class $\widetilde{h}$ belongs to $E_{m_g}$ and is orthogonal to G. One then has

for every $h'\in F$ of class $\widetilde{h}'$. In view of the assumption on F, this means that the measure $h(1 +|g|^2)\cdot \mu$ is zero, hence that $h$ is zero $\mu$-almost everywhere since $h$ is moderate.

Let $p$ be a real number $\geqslant 1$. Let $h$ be an element of $\mathscr{L}_K^{\infty}(X, \mu)$. The multiplication operator $\widetilde{m}_h$ by $h$ is an endomorphism of $L^p_K(X, \mu)$(IV, p. 186, No.$^o5$). Suppose that the set Y of the $x\in X$ such that $h(x) = 0$ is locally $\mu$-negligible. Then the endomorphism $\widetilde{m}_h$ is injective (Lemma 7 of IV, p. 186). Let us denote by $h^{-1}$ the function on X equal to 0 on Y and to $x\mapsto 1/h(x)$ on X- Y. The reciprocal partial operator $\widetilde{m}^{-1}_h$ is the multiplication operator by $h^{-1}$ from $L^p_K(X, \mu)$ into $L^p_K(X, \mu)$, that is to say, $\widetilde{m}^{-1}_h=\widetilde{m}_{h^{-1}}$. In fact, the image of $\widetilde{m}_h$ is the space of classes of functions $g\in \mathscr{L}_K^p(X, \mu)$ of the form $g=hf$ for $f\in \mathscr{L}_K^p(X, \mu)$. This condition is equivalent to $g(x)/h(x) =f(x)$ for every $x\in X$- Y and $g(x) = 0$ if $x\in Y$. This implies that the domain of $\widetilde{m}^{-1}_h$ in $L^p_K(X, \mu)$ is the domain of $\widetilde{m}_{h^{-1}}$, and that the equality $\widetilde{m}^{-1}_h=\widetilde{m}_{h^{-1}}$ is valid.

In what follows, we shall sometimes denote simply by $m_h$ the partial multiplication operator by $h$ from $L^{p^1}_K(X, \mu)$ into $L^{p_2}_K(X, \mu)$.