Let $E$ and $F$ be complex Banach spaces. We denote by $\overline{E}$ the compex conjugate of $E$, that is, the vector space $E$ with the same norm but with the conjugate multiplication by a complex scalar. The identity map defines a natural antilinear isomorphism $ x\to \overline{x}$ from $E$ onto $\overline{E}$. Recall that if $S\colon E\to F$ is a linear map we can define $\overline{S}\colon \overline{E} \to \overline{F}$ by $\overline{S}(\overline{x})=\overline{S(x)}$.
Let $H$ be a complex Hilbert space. Let $T\colon H \to E$ be a bounded linear map. Do we have $ \vert\vert T \vert\vert_{H \rightarrow E}^2=\vert\vert T\overline{T^*}\colon \overline{E^*} \rightarrow E\vert\vert\ ? $ where $T^*\colon E^* \to H^*$ is the adjoint map.