hyperbolic isomorphism
Let X be a Banach space and T:X→X a continuous
linear isomorphism. We say that T is an hyperbolic isomorphism if its spectrum is disjoint with the unit circle, i.e. σ(T)∩{z∈ℂ:|z|=1}=∅.
If this is the case, by the spectral theorem there is a
splitting of X into two invariant subspaces
, X=Es⊕Eu (and therefore, a corresponding splitting of T into two operators Ts:Es→Es and Tu:Eu→Eu, i.e. T=Ts⊕Tu), 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 T-1u there exists an equivalent
(box-type) norm ∥⋅∥1 such that
∥Ts∥1<λ and ∥T-1u∥1<λ |
and
∥x∥1=max{∥xu∥1,∥xs∥1}. |
In particular, λ can be chosen smaller than 1, so that Ts and T-1u 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 |