Gabriela Kohr

2. Univalent subordination chains in several complex variables
• There exist biholomorphic mappings on the unit ball B of an infinite
dimensional complex Banach space X which are not bounded on the
ball Br for r ∈ (0, 1).
• Graham, Hamada, K (2012): Let f (x) = (0, f2(x1), f3(x2), . . .) for
x = (x1, x2, . . .) ∈ ℓ2, where fn+1(xn) = 1
n(n+1) (2xn) n+1 , n ≥ 1. Also, let
F(x) = x + f (x). Then F is biholomorphic on ℓ2 and is not bounded on
Br for r ∈ (2/3, 1).
• L. Harris, S. Reich, D. Shoikhet (2000): Let X be a complex Banach
space and let h : B → X be a holomorphic mapping. If L(h) is finite,
then hs is bounded on B for each s ∈ (0, 1), where hs(z) = h(sz) and
L(h) = lim
s→1 sup Re V (hs).
• It is natural to consider to what extent such phenomena require
changes in the development of Loewner theory in complex Banach
spaces.
Gabriela Kohr (UBB Cluj) Geometric and analytic aspects of Loewner chains 14 / 62