Gabriela Kohr

3. Parametric representation on the unit ball
T. Poreda, 1991 (A < 2m(A)); M. Elin, S. Reich, D. Shoikhet,
2004 (k+(A) < 2k−(A)); Duren, Graham, Hamada, K., 2008.
Theorem
Let A ∈ L(C n , C n ) be an operator such that k+(A) < 2m(A). Also let
f : B n → C n be a spirallike mapping with respect to A and let
h(z) = [Df (z)] −1 Af (z) for z ∈ B n . Then f (z, t) = e tA f (z) is a univalent
subordination chain and
lim
t→∞ eAtv(z, s, t) = f (z, s)
locally uniformly on B n , where v(z, s, t) = f −1 (e (s−t)A f (z)). Also
∂f
∂t (z, t) = Df (z, t)h(z), t ≥ 0, z ∈ Bn .
Spirallike mappings w.r.t. A with k+(A) < 2m(A) have a
”generalized” parametric representation.
Gabriela Kohr (UBB Cluj) Geometric and analytic aspects of Loewner chains 37 / 62