Connected components in the space of composition operators on \(H^\infty\) functions of many variables (Q1865894)

Let \(H^\infty (B_E) \) be the space of all bounded analytic functions on the open unit ball \(B_E \) of a Banach space \(E . \) For an analytic mapping \( \phi :B_E \to B_F,\) the operator \( C_\phi : H^\infty (B_F)\to H^\infty (B_E) \) defined by \(C_\phi (f) =f\circ \phi \) is called the composition operator. For the case that \(E\) and \(F\) are the set of complex numbers, \textit{B. MacCluer, S. Ohno} and \textit{R. Zhao} have shown in [Integral Equ. Oper. Theory 40, 481-494 (2001; Zbl 1062.47511)] that (1) the identity mapping is an isolated point in the set \( C( H^\infty (B_F), H^\infty (B_E)) \) of all composition operators and (2) the path connected components in \( C( H^\infty (B_F), H^\infty (B_E))\) are open balls of radius 2 with respect to the operator norm. In the paper under review, the authors discuss the case of arbitrary Banach spaces \(E, F.\). In general, the set of all composition operators whose ranges lie strictly in \(B_E \) form a path connected component. For \(E=F=C_0 (X),\) the identity operator is an isolated point. It is conjectured that this should be true for general Banach spaces. If \(F\) is a Hilbert space or a \(C_0(X)\) space, two composition operators \( C_\phi , C_\psi \) are in the same path connected component if and only if \( ||C_\phi -C_\psi ||<2 \) with respect to the operator norm.