hyperbolic isomorphism

Let $X$ be a Banach space and $T:X\to X$ a continuous linear isomorphism. We say that $T$ is an hyperbolic isomorphism if its spectrum is disjoint with the unit circle, i.e. $\sigma (T)\cap \{z\in ℂ:|z|=1\}=\mathrm{\varnothing }$.

If this is the case, by the spectral theorem there is asplitting of $X$ into two invariant subspaces, $X=E^s\oplus E^u$ (and therefore, a corresponding splitting of $T$ into two operators $T^s:E^s\to E^s$ and $T_u:E^u\to E^u$, i.e. $T=T_s\oplus T_u$), such that and $\sigma (T_u)=\sigma (T)\cap \{z:|z|>1\}$. Also, for any $\lambda$ greater than the spectral radius of both $T_s$ and $T_u^{-1}$ there exists an equivalent (box-type) norm $\parallel \cdot {\parallel }_1$ such that

and

$${\parallel x\parallel }_1=\max \{{\parallel x_u\parallel }_1,{\parallel x_s\parallel }_1\}.$$

In particular, $\lambda$ can be chosen smaller than $1$, so that $T_s$ and $T_u^{-1}$ are contractions.