Given two hilbert spaces $X,Y$, and a bounded linear $T:X\to Y$, define $S:Y\to X$ by
$$ S=J_{X}^{-1} \circ T' \circ J_Y $$
Where $T':Y'\to X'$ is given by $T'(y')=y'\circ T$ for $y'\in Y'$ and $J_X ,J_Y$ are canonical isometries between their respective spaces and duals - That is, for example, $J_X(x)(y)=<x,y>$ for $x,y\in X$
Then one can check $<Tx,y>=<x,Sy>$ for all $x\in X$, $y\in Y$, except that I don't see how- could anyone explain it explicitly?
This can be done by a straightforward calculation: For all $x \in X$ and $y \in Y$ we have \begin{align*} \langle x, S(y) \rangle &= \langle x, J_X^{-1}(T'(J_Y(y))) \rangle = \langle x, J_X^{-1}(T'( \langle y, - \rangle)) \rangle = \langle x, J_X^{-1}(\langle y, T(-) \rangle) \rangle \\ &= \overline{\langle J_X^{-1}(\langle y, T(-) \rangle), x \rangle} = \overline{J_X(J_X^{-1}(\langle y, T(-) \rangle))(x)} = \overline{\langle y, T(-) \rangle(x)} \\ &= \overline{\langle y, T(x)} = \langle T(x), y \rangle. \end{align*}