Hyperrigid generators in C*-algebras

In this article, we show that, if $S\in \mathcal{B}(H)$ is irreducible and essential unitary, then $\{S,SS^*\}$ is a hyperrigid generator for the unital $C^*$-algebra $\mathcal{T}$ generated by $\{S,SS^*\}$. We prove that, if $T$ is an operator in $\mathcal{B}(H)$ that generates an unital $C^*$-algebra $\mathcal{A}$ then $\{T,T^*T,TT^*\}$ is a hyperrigid generator for $\mathcal{A}$. As a corollary it follows that, if $T\in \mathcal{B}(H)$ is normal then $\{T,TT^*\}$ is hyperrigid generator for the unital $C^*$-algebra generated by $T$ and if $T\in \mathcal{B}(H)$ is unitary then $\{T\}$ is hyperrigid generator for the $C^*$-algebra generated by $T$. We show that if $V\in \mathcal{B}(H)$ is an isometry (not unitary) that generates the $C^*$-algebra $\mathcal{A}$ then the minimal generating set $\{V\}$ is not hyperrigid for $\mathcal{A}$.