Assume $(H,(\cdot,\cdot)_H)$, and $(G,(\cdot,\cdot)_G)$ are two vector spaces with inner product. Suppose $A:H\rightarrow G$ is a linear isometry.
Let $T(H)_{*}$ be the completion of $T(H)$ with respect to $(\cdot,\cdot)_G$.
Let $H_{*}$ be the completion of $H$ with respect to $(\cdot,\cdot)_H$.
Question: Is it true that $T(H)_{*}=T(H_{*})$, where $T$ has been extended by continuity to $H_{*}$?
Yes, it is true.
To prove $T(H_*)\subseteq T(H)_*$, you should prove that the image of every Cauchy sequence $\{x_n\}$ in $H$ is a Cauchy sequence in $G$ and the limit of $\{x_n\}$ is mapped to the limit of $\{Tx_n\}$.
To prove $T(H)_*\subseteq T(H_*)$, you should prove that the inverse image of every Cauchy sequence in $G$ is a Cauchy sequence in $H$ whose limit in $H$ is the inverse image of the limit of the former sequence in $G$.