Multiplication operators on the Bloch space of infinite dimensional bounded symmetric domains (Q2064061)

Given a bounded symmetric domain realized as the unit ball \(\mathbb{B}_V\) of a JB*-triple \(V\), the authors investigate the multiplication operators \(M_\psi(f) := \psi f\) defined for functions \(f\) in the Bloch spaces \(\mathcal{B}(\mathbb{B}_V)\). They show that \(M_\psi\) with a holomorphic function \(\psi:\mathbb{B}_V\to\mathbb{C}\) is bounded with respect to the natural operator norm of \(\mathcal{B}(\mathbb{B}_V)\) if and only if \(\psi\) is bounded and \(\sigma_\psi < \infty\) for \(\sigma_\psi := \sup_{z\in\mathbb{B}_V} \omega(z)Q_\psi(z)\) where \(\omega(z) :=\sup\big\{ \vert f(z)\vert : f(0)=0, \Vert f\Vert_{\mathcal{B}(\mathbb{B}_V)} \le 1 \big\}\) and \(Q_\psi(z) := \sup_{0\ne v \in V} \kappa(z,v)^{-1} \vert D\psi(z)v \vert\) denotes the norm of the Fréchet derivative of \(\psi\) at the point \(z\) with respect to the infinitesimal Kobayashi metric. In the case \(\Vert M_\psi \Vert_{\mathcal{B}(\mathbb{B}_V)} <\infty\), its spectrum has the form \(\overline{\psi(\mathbb{B}_V)}\), furthermore, if in addition \(\Vert \psi\Vert_\infty <1\), then \(M_\psi\) is also power bounded and uniformly mean ergodic.