Is this true? If $\varphi\in \mathcal{B}_{\mathcal{A}}(E,F)$ and preserving density, then $\varphi$ is surjection, why?. Such that $E$ and $F$ are right Hilbert $\mathcal{A}$-modules and $\mathcal{B}_{\mathcal{A}}(E,F)$ is the set of all bounded $\mathcal{A}$-linear operator from $E$ to $F$. ( $\varphi :E\longrightarrow F$ is called $\mathcal{A}$-linear if it is linear and for each $a\in \mathcal{A}$ and $x\in E$, $\varphi(xa)=\varphi(x)a$ ). Also we say that $\varphi\in \mathcal{B}_{\mathcal{A}}(E,F)$ preserves density on $E$ if the subset $B$ of $E$ is dense in $E$, then $\varphi (B)$ is dense in $F$.
My answer: Let $A=\mathbb{C}, E=F=L^2[0,1]$ and $(\varphi f)x=\int_{0}^{x}f(t)dt$. Is it true?