# 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. $\sigma (T)\cap \{z\in \u2102:|z|=1\}=\mathrm{\varnothing}$.

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 $$ 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

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

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

Title | hyperbolic isomorphism |
---|---|

Canonical name | HyperbolicIsomorphism |

Date of creation | 2013-03-22 13:39:34 |

Last modified on | 2013-03-22 13:39:34 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 10 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 37D05 |

Classification | msc 46B03 |

Synonym | linear hyperbolic isomorphism |