Some problems in operator theory on bounded symmetric domains

We discuss some problems concerning the operators Km(Z, Z*), where Km is the reproducing kernel of the Peter-Weyl space with signature m and Z stands for the commuting tuple of operators of multiplication by the coordinate variables on a bounded symmetric domain. These problems have applications for construction of operator models, as well as for a possible generalization of the recent dilation theory for row contractions due to Arveson.