Nonlinear resolvent of holomorphic generators

יום ב', 07/01/2019 - 14:00
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This talk is based on joint work with Mark Elin and Toshiyuki Sugawa. Let $f$
\ be the infinitesimal generator of a one-parameter semigroup $\left\{
F_{t}\right\} _{t>0}$ of holomorphic self-mappings of the open unit disk,
i.e., $f=\lim_{t\rightarrow 0}\frac{1}{t}\left( I-F_{t}\right) .$ In this
work, we study properties of the resolvent family $R=\left\{ \left(
I+rf\right) ^{-1}\right\} _{r>0}$ \ in the spirit of geometric function
theory. We discovered, in particular, that $R$ forms an inverse Loewner
chain and consists of starlike functions of order $\alpha >1/2$. Moreover,
each element of $R$ satisfies the Noshiro-Warshawskii condition $\left(
\func{Re}\left[ \left( I+rf\right) ^{-1}\right] ^{\prime }\left( z\right)
>0\right) .$ This, in turn, implies that all elements of $R$ are also
holomorphic generators. Finally, we study the existence of repelling fixed
points of this family.