Hilbert space adjoint vs Banach space adjoint?

$\newcommand{\hdual}{\mathsf H}$Let $T^\hdual$ denote the "Hilbertian" adjoint (usually called conjugate transpose) and $T^*$ the "Dual" or "Banachian" adjoint. Let $\mathrm E_X:X \to X^*$ be the natural embedding of a Hilbert space onto its dual, namely $x \mapsto \langle x,\cdot \rangle$. Here the inner product is taken to be linear in the second component and conjugate-linear in the first.

We can show that $$ T^\hdual = \mathrm E_X^{-1} T^* \mathrm E_Y^{\vphantom{-1}} $$