Fredholm composition operators on algebras of analytic functions on Banach spaces (Q849004)

Let \(E\) be a complex Banach space with open unit ball \(B_E\), and let \(\varphi:B_E\rightarrow B_E\) be a nonconstant analytic map. The composition operator \(C_\varphi\) is defined by \(C_\varphi(f)=f\circ \varphi\) and acts on the uniform algebra of bounded analytic functions on \(B_E\) endowed with the norm \(\| f\|=\sup_{x\in B_E}|f(x)|\). Under the additional assumption that \(E\) possesses the approximation property, it is proved that Fredholm composition operators are invertible and arise from analytic automorphisms of the ball.