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. $\sigma(T)\cap\{z\in\mathbb{C}:|z|=1\}=\emptyset$.

If this is the case, by the spectral theorem there is a splitting 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 $\sigma(T_{s})=\sigma(T)\cap\{z:|z|<1\}$ 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 $\|\cdot\|_{1}$ such that

 $\|T_{s}\|_{1}<\lambda\textnormal{ and }\|T_{u}^{-1}\|_{1}<\lambda$

and

 $\|x\|_{1}=\max\{\|x_{u}\|_{1},\|x_{s}\|_{1}\}.$

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

Title hyperbolic isomorphism HyperbolicIsomorphism 2013-03-22 13:39:34 2013-03-22 13:39:34 Koro (127) Koro (127) 10 Koro (127) Definition msc 37D05 msc 46B03 linear hyperbolic isomorphism