On the structure of $C_0$-semigroups of holomorphic Carath\'eodory isometries


We extend Vesentini's description of the
infinitesimal generators of strongly continuous one-parameter semigroups 
of holomorphic Carat\'oodory isometries
of the unit ball of a complex Hilbert space to the setting of 
reflexive Cartan factors.
Our treatment is based on intensive use of joint fixed points
along with Kaup type ideas with partial vector fields of second degree.
In particular we establish closed formulas for the Hilbert space case
in terms of spectral resolutions of skew self-adjoint dilations 
related to the Reich-Shoikhet non-linear infinitesimal generator.
We also provide partial results toward a Hille-Yosida type theory
for holomorphic self-maps of bounded domains in Banach spaces.