1. AND CONTINUOUS SEMIGROUPS
ABEL AVERAGES
OF DISCRETE
AND CONTINUOUS SEMIGROUPS
David Shoikhet
ORT Braude College, Karmiel

13. Fixed points of holomorphic mappings in Banach and Hilbert Spaces
In general Banach spaces holomorphic mappings not necessarily have fixed points even the underlined domain is bounded.
X = c0 ={( x1, x2,...) : |xn|→0}, D = {||x||<1}, F(x) = (½, x1, x2,...)
Then, but F has no fixed point in X.
Example 14

14. Fixed points : existence
Theorem
Let D be a nonempty domain in a complex Banach space X and let h : D → D be a bounded holomorphic mapping.
If h(D) lies strictly inside D, then h has a unique fixed point in D.
(Earle-Hamilton, 1970) τ 15

15. The structure of fixed point sets
Let B be the open unit ball in a complex Hilbert space
Let and assume that
Then is an affine manifold in B.
Theorem
W.Rudin, 1978
Moreover, if τ = 0 and A=F ’(0), then FixB(F) = FixB(A).
In particular, if A = I then F=I (Cartans’ uniqueness theorem)
Let be a self-mapping of a bounded convex domain in a reflexive Banach space X.
Then if it is a holomorphic retract of D.
Theorem
J. P.Vigue, 1986,
D. Shoikhet,1986 16

18. .

29. Thank you
for your attention