In this question Operators on Hilbert $C^*$-module and families of Fredholm operators I asked about the relation between being a family of compact operators $F:X \to K(H)$ on Hilbert space $H=\ell^2$ and being $C(X)$-compact operator on Hilbert module $H_{C(X)}$ (here $X$ is a compact space). If I take a family $F$ like above then I can view it as an operator on $H_{C(X)}$ by the formula $F(\xi)(x)=F_x(\xi(x))$ where $\xi \in H_{C(X)}$ and therefore $\xi(X) \in H=\ell^2$ so $F_x(\xi(x))$ does makes sense. I wonder whether the opposite is true, namely if we have a $C(X)$-compact operator on $H_{C(X)}$ may we somehow interpret it as a continuous family of compact operators? I suspect that this no longer will be true for *all* operators, so in the case if the answer is affirmative for compact operators, and negative for general operators, I would also like to understand where the obstruction lies.

## 1 Answer

The Hilbert module $H_{C(X)}$ is the space of all continuous functions $X \to H$, and hence for each $s \in H_{C(X)}$, we can talk about $s(x) \in H$, for each $x \in X$. Assume that $F:H_{C(X)} \to H_{C(X)}$ is $C(X)$-linear. We get individual operators $F_x: H \to H$, namely $$F_x (v):= (F\tilde{v})(x), $$ where $v \in H$ yields the constant function $\tilde{v}: X \to H$. It is then not hard to show that for each $s \in H_{C(X)}$ and $x \in X$, the relation $$(Fs)(x)= F_x (s(x))$$ holds. Claim: if $F$ is $C(X)$-compact, then $F_x$ is compact for each $x \in X$ and $x \mapsto F_x$ is (norm-)continuous. Proof: let $\epsilon>0$. Then, by the definition of a $C(X)$-compact operator, there exists finitely many $s_i, t_i \in H_{C(X)}$, such that the operator $$G=\sum_i t_i \langle s_i, \_ \rangle$$ on $H_{C(X)}$ satisfies $\| G-F\| \leq \epsilon$. It is clear that $G_x$ is compact for each $ x\in X$ and $x\mapsto G_x$ is norm-continuous. Note that $\| F_x - G_x\| \leq \epsilon$. Since $\epsilon$ was arbitrary, the proof is complete.

WARNING: if $F$ is a $C(X)$-Fredholm operator on $H_{C(X)}$, then you also get Fredholm operators $F_x$ on $H$, but the map $x\mapsto F_x$ is not continuous, at least not if the target has the norm topology.