Let $H$ be an Hilbert space over $\mathbb{C}$
Let $\{x_m\}_{m \in \mathbb{N}} \in H$ and $\{y_m\}_{m \in \mathbb{N}} \in H$ be two basic sequences in $H$
Let $X_m = \overline{\operatorname{span}} \{x_p\}_{p \geq m}$
Let $Y_m = \overline{\operatorname{span}} \{y_p\}_{p \geq m}$
We know that there is an isomorphism $T : \overline{\operatorname{span}} \{x_m\}_{m \in \mathbb{N}} \to \overline{\operatorname{span}} \{y_m\}_{m \in \mathbb{N}}$ such that $Tx_m = y_m$
Is it true that:
$$ \bigcap_{m=1}^\infty \left( X_m + Y_m \right) =\{0\} $$
Thanks.