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hyperbolic isomorphism (Definition)

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}:\vert z\vert=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:\vert z\vert<1\}$ and $ \sigma(T_u)=\sigma(T)\cap\{z:\vert z\vert>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 $ \Vert\cdot\Vert _1$ such that

$\displaystyle \Vert T_s\Vert _1 < \lambda \textnormal{ and } \Vert T_u^{-1}\Vert _1 < \lambda$
and
$\displaystyle \Vert x\Vert _1 = \max\{\Vert x_u\Vert _1,\Vert x_s\Vert _1\}.$
In particular, $ \lambda$ can be chosen smaller than $ 1$, so that $ T_s$ and $ T_u^{-1}$ are contractions.



"hyperbolic isomorphism" is owned by Koro.
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Other names:  linear hyperbolic isomorphism
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Cross-references: contractions, norm, equivalent, spectral radius, operators, invariant subspaces, spectral theorem, unit circle, disjoint, spectrum, linear isomorphism, continuous, Banach space
There are 2 references to this entry.

This is version 7 of hyperbolic isomorphism, born on 2003-05-29, modified 2007-11-30.
Object id is 4315, canonical name is HyperbolicIsomorphism.
Accessed 3143 times total.

Classification:
AMS MSC: 37D05 (Dynamical systems and ergodic theory :: Dynamical systems with hyperbolic behavior :: Hyperbolic orbits and sets)
 46B03 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Isomorphic theory of Banach spaces)
Pending Errata and Addenda
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