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ABEL AVERAGES OF DISCRETE AND CONTINUOUS SEMIGROUPS

Explore the existence and uniqueness of fixed points in bounded holomorphic mappings in Banach spaces. Learn about affine manifolds and fixed point sets in complex Hilbert spaces. Examples and in-depth theorems provided.

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ABEL AVERAGES OF DISCRETE AND CONTINUOUS SEMIGROUPS

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

  2. 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

  3. 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

  4. 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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  6. Thank you for your attention

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