Let $\mathcal{H}=L^2([0,1])\oplus L^2([0,1])$ and $T=M_g\oplus M_g$ where $g(t)=t$ and $(M_gf)(t)=g(t)f(t)$. I want to verify that $\mathcal{H}$ has no cyclic vector for $T$.
The operator norm closure of all polynomials in $T$ is $\{M_\varphi\oplus M_\varphi:\varphi\in C([0,1])\}$. Suppose $(f,g)\in\mathcal{H}$ is cyclic for $T$, I want to show that $\{(\varphi f,\varphi g): \varphi\in C([0,1])\}$ is not dense in $\mathcal{H}$. How do I proceed?
Edit: Is the following correct?
Suppose $\mathcal{A}=\{(\varphi f,\varphi g): \varphi\in C([0,1])\}$ is dense in $\mathcal{H}$. Then $\mathcal{A}^\perp=\{0\}$. However $(\bar{g},-\bar{f})\in\mathcal{A}^\perp$ and hence $f$ and $g$ are zero almost everywhere, which clearly contradicts that $\mathcal{A}$ is dense in $\mathcal{H}$.