Let $H$ be a separable, complex Hilbert space and let $\mathcal{B}(H)$ denote the algebra of bounded linear operators on $H$. Let $T \in \mathcal{B}(H)$. We define $$ A = \{ p(T,T^*) : p \in \mathbb{C}[z_1,z_2] \}.$$ $A$ is the subalgebra generated by $T,T^*$ - or the $*$-algebra generated by $T$. If we denote by $\overline{A}^\sigma$ its weak (operator) closure, I would like to know whether can the equality $$\overline{A}^\sigma = \mathcal{B}(H)$$ holds.

Thank you very much.