Suppose $Y$ is a Banach space and $X$ is a **finite-dimensional** subspace of $Y$. Further assume $T:X \rightarrow X$ is a linear operator which is power bounded from above **and below**, in other words there is $0 < c \leq 1 \leq C < \infty$ such that $c \cdot \|x\| \leq \|T^n(x)\| \leq C \cdot\|x\|$ for all $n \in \mathbb{N}$ and all $x \in X$.

Can $T$ be extended to a linear operator $\tilde{T}:Y \rightarrow Y$ which satisfies the same power bounds?

If it is not true in general, are there nice conditions which make it true? (It is important to my application that there is a lower bound.)

*I apologize in advance if this is too trivial for Mathoverflow. I tried to look it up but couldn't find the answer. The impression I got is that extending linear operators is not as well known as extending linear functionals (Hahn-Banach). Also, while this is for an analysis paper, I am not an analyst. Hence I am not always sure what is common knowledge and what isn't. Thanks!*