hyperbolic isomorphism

Let X be a Banach spaceMathworldPlanetmath and T:XX a continuousMathworldPlanetmathPlanetmath linear isomorphism. We say that T is an hyperbolic isomorphism if its spectrum is disjoint with the unit circle, i.e. σ(T){z:|z|=1}=.

If this is the case, by the spectral theoremMathworldPlanetmathPlanetmath there is a splitting of X into two invariant subspacesPlanetmathPlanetmath, X=EsEu (and therefore, a corresponding splitting of T into two operators Ts:EsEs and Tu:EuEu, i.e. T=TsTu), such that σ(Ts)=σ(T){z:|z|<1} and σ(Tu)=σ(T){z:|z|>1}. Also, for any λ greater than the spectral radius of both Ts and Tu-1 there exists an equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath (box-type) norm 1 such that

Ts1<λ and Tu-11<λ



In particular, λ can be chosen smaller than 1, so that Ts and Tu-1 are contractionsPlanetmathPlanetmath.

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