Page 159 - Textos de Matemática Vol. 39

STABILITY OF INVARIANT SUBSPACES 149
Definition 1.2. X ∈ M∗n,d is said to be A-stable if the following conditions are satisfied:
(i) X ∈ S(A);
(ii) ∀ε>0∃δ>0suchthatif∥A−A′∥<δthen∃X′∈S(A′)with
∥X−X′ ∥<ε.
Theorem 1.3. For a matrix A ∈ Cn×n, X ∈ M∗n,d is A-stable if and only if
< X > is A-stable.
Proof. Assume that X is A-stable. By (i) of Definition 1.2 and Proposition 1.1, < X > is A-invariant. Fix ε > 0. As π is a continuous map, ∃ η > 0 such thatif∥X−X′ ∥<ηthenθ(<X>,<X′ >)<ε.SinceXisA-stable,forthis η,∃δ>0suchthatif∥A−A′∥<δ,thereexistsX′ ∈S(A′)with ∥X−X′∥<η. Putting all together we have that for the given ε > 0 ∃ δ > 0 such that if ∥A−A′∥ < δ then there exists X′ ∈ S(A′) such that θ(< X >,< X′ >) < ε. Hence, < X′ > is A′-invariant and < X > is A-stable.
Conversely, assume that V ∈ Grd(Cn) is A-stable, and let X ∈ M∗n,d be abasismatrixofV;i.e.V=<X>.Fixε>0.SinceM∗n,d isopen,thereis ε′ >0suchthatε≥ε′ andBε′(X)⊂M∗n,d.Now,πisanopenmapandso π(Bε′(X)) ⊂ Grd(Cn) is open. Therefore, as V ∈ π(Bε′(X)), there is η > 0 suchthatifθ(V,V′)<ηthen∃X′ ∈M∗n,d suchthat∥X−X′ ∥<ε′ ≤εand <X′ >=V′.AsVisA-stable,forthisη>0∃δ>0suchthatif∥A−A′ ∥<δ then there exists V′ ∈ Grd(Cn), A′-invariant, such that θ(V,V′) < η. Thus, ∃X′ ∈M∗n,d suchthat∥X−X′ ∥<εand<X′ >=V′.AsV′ isA′-invariant we conclude that X′ ∈ S(A′) and ∥ X −X′ ∥< ε. This proves that X is A-stable.
The rest of the paper is organized following the same methodology as in [3]. In Section 2 we prove that it is enough to study the case when A is nilpotent and in Section 3 we characterize the invariant subspaces of nilpotent matrix which are stable.
2. The reduction to the nilpotent case
The difficult task in characterizing the stable subspaces of a square matrix is the reduction to the case when a matrix has only one eigenvalue. Once this is done it is easy to prove that such an eigenvalue can be taken to be zero; i.e. the matrix is nilpotent. In fact we have the following result
Proposition 2.1. If A ∈ Cn×n and λ0 ∈ C then X ∈ M∗n,d is A-stable if and only if it is (λ0In − A)-stable.
Proof. It suffices to prove that if X is A-stable then it is (λ0In − A)-stable for any λ0 ∈ C. In fact, once this is proved, the converse is a consequence of the following identity: A = λ0In − (λ0In − A).