1. 11 M. Elin The Galilee Research Center for Applied Mathematics of ORT Braude College Karmiel, Israel

2. Continuous Semigroups

3. 33 Spirallike and starlike mappings

4. 44 In the one-dimensional settings - the well-known criteria of Nevanlinna, Study and Špaček In multi-dimensional situations – Suffridge, Gurganus, Pfaltzgraff, Gong, … In multi-dimensional situations – not all of the analogues hold, proofs are very complicated, examples are rather hard to construct. Spirallike and starlike mappings

5. 55 Since the work of Roper and Suffridge in 1995, there has been considerable interest in constructing holomorphic mappings of the unit ball in a Banach space with various geometric properties by using mappings with similar properties acting in a subspace. Such properties include convexity, starlikeness, spirallikeness, and so on. It is also of interest to extend subordination chains, semigroups and semigroup generators. Extension Operators

6. Roper-Suffridge extension operator

7. 7 Modifications of R-S extension operator Pfaltzgraff, Suffridge, 1997 I.Graham, G.Kohr, M.Kohr, 2000 I.Graham, G.Kohr, 2000

8. 8 Modifications of R-S extension operator - the chain rule is invertible

9. 99 Some notations

10. 10 Main notation and notion

11. 11 Extension operators for semigroups

12. 12 Extensions of spirallike mappings and subordination chains

13. 13 Extensions of spirallike mappings

14. 14 Extensions of spirallike mappings

15. 15 Extensions of spirallike mappings

16. 16 Extension operator: Extreme Points, Support Points Further question

17. Muir’s extension operator

18. Geometric explanation

20. Covering results

21. Roper-Suffridge type operator

22. Spiralikeness for Muir’s type operator

23. 23 Thank you for your attention!