ABEL AVERAGES OF DISCRETE AND CONTINUOUS SEMIGROUPS

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

13. Example X = c0 ={( x1, x2,...) : |xn|→0}, D= {||x||<1}, F(x) = (½, x1, x2,...) Then, but F has no fixed point in X. 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. 14

14. Fixed points : existence Theorem LetD be a nonempty domain in a complex Banach spaceXand leth:D → D be a bounded holomorphic mapping. Ifh(D) lies strictly insideD, then h has a unique fixed point inD. (Earle-Hamilton, 1970) τ 15

15. Theorem Let and assume that Then is an affine manifold in B. 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. The structure of fixed point sets Let B be the open unit ball in a complex Hilbert space W.Rudin, 1978 Moreover, ifτ = 0andA=F’(0),thenFixB(F) = FixB(A). In particular, if A = Ithen F=I (Cartans’ uniqueness theorem) J. P.Vigue, 1986, D. Shoikhet,1986 16

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29. Thank you for your attention