# Nonlinear resolvent of holomorphic generators

Seminar
Speaker
Prof. David Shoikhet, Holon Institute of Technology, Israel
Date
07/01/2019 - 15:45 - 14:00Add to Calendar 2019-01-07 14:00:00 2019-01-07 15:45:00 Nonlinear resolvent of holomorphic generators 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. 2nd floor Colloquium Room, Building 216 אוניברסיטת בר-אילן - המחלקה למתמטיקה mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
2nd floor Colloquium Room, Building 216
Abstract

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.

תאריך עדכון אחרון : 06/01/2019