Norm of the multiplication operators from [formula omitted] to the Bloch space of a bounded symmetric domain

Abstract

Let D be a bounded symmetric domain in C N and let ψ be a complex-valued holomorphic function on D. In this work, we determine the operator norm of the bounded multiplication operator with symbol ψ from the space of bounded holomorphic functions on D to the Bloch space of D when ψ fixes the origin. If no restriction is imposed on the symbol ψ, we have a formula for the operator norm when D is the unit ball or has the unit disk as a factor. The proof of this result for the latter case makes use of a minimum principle for multiply superharmonic functions, which we prove in this work. We also show that there are no isometries among the multiplication operators when the domain does not have exceptional factors or the symbol fixes the origin.