Let $E$ be a Hilbert module over a $C^*$-algebra $A$. Let $T\colon E\to E$ be a densely defined, unbounded $A$-linear operator. (In particular, the initial domain of $T$ is an $A$-submodule of $E$.) If the operators $T\pm i$ each has dense range, then $T$ is an essentially self-adjoint, regular operator on $E$, and there is a continuous functional calculus

$$F_T\colon C_b(\mathbb{R})\to\mathcal{B}_A(E),$$

where $C_b(\mathbb{R})$ denotes the bounded continuous $\mathbb{C}$-valued functions on $\mathbb{R}$, and $\mathcal{B}_A(E)$ denotes the $C^*$-algebra of bounded adjointable $A$-linear operators on $E$.

**Question:** Suppose $D(T)$ is not an $A$-submodule of $E$, but only an $A_0$-submodule, where $A_0$ is a (metrically) dense $*$-subalgebra of $A$. Then, still assuming that $T\pm i$ each has dense range, is it possible to construct a continuous functional calculus for $T$?

**Comment:** It seems to me that even though in this situation $T$ is only $A_0$-linear, the operator $1+T^*T$ is injective, and that $(1+T^*T)^{-1}$ extends to an $A$-linear operator. Similarly, it seems possible to make sense of the operator $T(1+T^*T)^{-1/2}$ as being $A$-linear.