Let $\mathcal{H}$ be a Hilbert space and $T$ is a surjective bounded linear operator on $\mathcal{H}$. Suppose that $\mathcal{K}$ is a subspace of $\mathcal{H}$ such that $T|_{\mathcal{K}}$ is surjective. Does the relation $T^*\mathcal{H} \leq \mathcal{K}$ hold?
My attempt: Suppose that $h\in \mathcal{H}$. There are elements and $h_1\in \mathcal{K}$ and $h_2\in T^*\mathcal{H}$ such that $$Th_1=h=Th_2.$$ Thus $h_1-h_2\in \ker T$ and $h_1= (h_1-h_2)+h_2$ which yields that $T^*\mathcal{H} \leq \mathcal{K}$.
Is this argument correct?
Fix an orthonormal basis $\{e_n\}$, then define $T$ by $$ Te_n=\begin{cases} e_k,&\ n=2k\\ e_k,&\ n=2k-1 \end{cases} $$ Let $K=\operatorname{span}\{e_{2k}:\ k\in\mathbb N\}$. Then $T|_K$ is surjective. We have $$ \langle T^*e_k,e_{2j}\rangle=\langle e_k,Te_{2j}\rangle=\langle e_k,e_j\rangle=\delta_{k,j}=\langle e_{2k-1}+e_{2k},e_{2j}\rangle, $$ and $$ \langle T^*e_k,e_{2j-1}\rangle=\langle e_k,Te_{2j-1}\rangle=\langle e_k,e_j\rangle=\delta_{k,j}=\langle e_{2k-1}+e_{2k},e_{2j-1}\rangle, $$ so $$ T^*e_k=e_{2k-1}+e_{2k}. $$ In particular, $T^*e_1=e_1+e_2\not\in K$.