Gabriela Kohr

2. Univalent subordination chains in several complex variables
Definition
Let k+(A) = max{Re λ : λ ∈ σ(A)} be the upper exponential index
(Lyapunov) index of A, where σ(A) is the spectrum of A ∈ L(X).
Let k−(A) = min{Re λ : λ ∈ σ(A)} be the lower exponential index
(Lyapunov) index of A.
Remark
Let A ∈ L(X) be such that m(A) > 0. Then
(i) k−(A) ≤ m(A) ≤ k+(A) ≤ k(A) ≤ |V (A)| ≤ A ≤ e|V (A)|;
;
(ii) For each ω > k+(A), there is δ = δ(ω) > 0 s.t. etA ≤ δeωt , t ≥ 0;
(iii)
e m(A)t ≤ e tA (u) ≤ e k(A)t , t ∈ [0, ∞), u = 1.
k+(A) = limt→∞ ln etA t
(iv) L. Harris, 1971: If Pm : X → X is a homogeneous polynomial
mapping of degree m, then Pm ≤ km|V (Pm)|, where km = m m/(m−1)
when m ≥ 2 and k1 = e (k1 = 2 for complex Hilbert spaces).
Gabriela Kohr (UBB Cluj) Geometric and analytic aspects of Loewner chains 18 / 62