Executive Summary : | The study aims to investigate the commuting tuple of multiplication operators homogeneous under compact group defined on a reproducing kernel Hilbert space of vector valued holomorphic functions. The homogeneity of the multiplication tuple is equivalent to a transformation rule for the reproducing kernel. The main goal is to set up the machinery in a bounded symmetric domain and the maximal compact subgroup of its bi-holomorphic automorphism group. A detailed study of the commuting tuple of multiplication operators homogeneous under compact group is underway, with partial results obtained in the group of unitary matrices. A model theorem is proposed, stating that the reproducing kernel need not be invariant under the action of the compact group, but rather quasi-invariant. The study also investigates the joint hyponormality and subnormality of the commuting tuple of operators homogeneous under the compact group. |