Holomorphic Mappings and Their Fixed Points on Spin Factors

Abstract In this paper we study holomorphic properties of infinite dimensional spin factors. Among the infinite dimensional Banach spaces with homogeneous open unit balls, we show that the spin factors are natural outlier spaces in which to ask the question (as was proved in the early 1970s for Hilbert spaces): Do biholomorphic automorphisms g of the open unit ball B have fixed points in $$\overline{B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> ? In this paper, for infinite dimensional spin factors, we provide reasonable conditions on g that allow us to explicitly construct fixed points of g lying on $$\partial B$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>B</mml:mi> </mml:mrow> </mml:math> . En route, we also prove that every spin factor has the density property. In another direction, we focus on (compact) holomorphic maps $$f:B\rightarrow B$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>B</mml:mi> <mml:mo>→</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:math> , having no fixed point in B and examine the sequence of iterates $$(f^n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . As $$(f^n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> does not generally converge, we instead trace the target set T ( f ) of f , that is, the images of all accumulation points of $$(f^n)_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> </mml:math> , for any topology finer than the topology of pointwise convergence on B. We prove for a spin factor that T ( f ) lies on the boundary of a single bidisc unique to f .