Assume $T\in L(X)$ is a bounded linear operator on a Banach space $X$, and that $T_n$ is a restriction of $T$ on $R(T^n)=T^n(X)$. In general, almost nothing can be said concerning the relationship between the spectral properties of $T$ and $T_n$. However, under some conditions, it is shown that several spectral properties introduced recently are the same for $T$ and $T_n$.
"Spectral properties and restrictions of bounded linear operators." Ann. Funct. Anal. 6 (2) 173 - 183, 2015. https://doi.org/10.15352/afa/06-2-15