Adjoint operator between Hilbert spaces, does it map a subspace onto a subspace?

Question

Let $X$ and $Y$ be Hilbert spaces and let $f\colon X \to Y$ be a linear continuous bijection. Define the adjoint $f'\colon Y \to X$ by $(f'y, x)_X = (y, fx)_Y$.

Let $X_0$ and $Y_0$ be subspaces of $X$ and $Y$ respectively, and suppose that $f({X_0}) = Y_0$.

Does it follow that $f'|_{Y_0} \subset X_0$?

So does the adjoint map the subspace to the subspace if $f$ does?

Answer 1

Sometimes is good to look at the simple examples first. Let $X=Y=\mathbb C^2$, and $$ f=\begin{bmatrix}1&1\\0&1\end{bmatrix}. $$ Take $X_0=\left\{\begin{bmatrix}x\\0\end{bmatrix}:\ x\in\mathbb C\right\}$, and $Y_0=f(X_0)=X_0$. Now you can check that $$ f'(Y_0)=\left\{\begin{bmatrix}x\\ x\end{bmatrix}:\ x\in\mathbb C\right\}\not\subset X_0. $$