If $H_1$ and $H_2$ are Hilbert spaces and $T:H_1\rightarrow H_2$ a bounded linear operator, then one can show that $T$ is invertible if and only if there exists a constant $\alpha > 0$ such that $T^*T\geq\alpha I_{H_1}$ and $TT^*\geq\alpha I_{H_2}$.
My query is: if $H_1$ and $H_2$ are now Hilbert $C^*$-modules over some (not necessarily unital) $C^*$-algebra $A$, and $T$ is an adjointable operator $H_1\rightarrow H_2$, is it true that
1) $T^*T$ and $TT^*$ being positive elements in $\text{End}(H_1)$ and $\text{End}(H_2)$ respectively is equivalent to $T$ being invertible?
2) If the answer to 1) is no, is there another analogue of the Hilbert space result that is true for Hilbert modules?
Thanks.
Since the adjointable operators on $H_1$ (or $H_2$) form a C$^*$-algebra, one can reason the same as in the Hilbert space case.
Namely, if $T^*T\geq\alpha I_{H_1}$, then $T^*T$ is invertible, so there exists $X\in \text{End}(H_1)$ with $XT^*T=I_{H_1}$. Similarly, if $TT^*\geq\alpha I_{H_2}$ then there exists $Y$ with $TT^*Y=I_{H_2}$. So $T$ is invertible.
Conversely, if $T$ is invertible, let $Z$ be its inverse. Then $$T^*TZZ^*=T^*Z^*=(ZT)^*=I_{H_1}^*=I_{H_1}.$$ Similarly, $ZZ^*T^*T=I_{H_1}$. As $T^*T$ is positive and invertible in the C$^*$-algebra $\text{End}(H_1)$, there exists $\beta>0$ with $T^*T\geq\beta I_{H_1}$. Repeating the argument now for $T^*$, there exists $\gamma>0$ with $TT^*\geq\gamma I_{H_2}$. Now take $\alpha=\min\{\beta,\gamma\}$.