Research Projects

The study of homogeneous tuples of commuting bounded linear operators has significantly advanced over the years. Cowen and Douglas introduced the influential class $B_n(\Omega)$, and Misra classified homogeneous operators in $B_1(\mathbb{D})$. Wilkins extended this work to $B_2(\mathbb{D})$, while Misra and Korányi later provided a comprehensive classification of homogeneous operators in $B_n(\mathbb{D})$ using complex geometry and representation theory. Homogeneous operators on bounded symmetric domains have also been studied extensively. Chavan and Yakubovich proved that every U(d)-homogeneous operator tuple T with joint kernel of the adjoint of T being one dimensional cyclic subspace for T is unitarily equivalent to the d-tuple M of multiplication operator by the co-ordinate functions on a reproducing Kernel Hilbert space determined by a U(d)-invariant kernel. In this project, we try to investigate a structure theorem for the U(d)-homogeneous subnormal tuple of operators. Note that here we are not imposing any conditions on the dimension of the joint kernel of the adjoint operator tuple. In a recent work of Misra et.al. a concrete model for a commuting tuple N of G-homogeneous normal operators whose joint spectrum is precisely the closure of G- orbits, has been developed. Here G is a locally compact second countable group. In fact, it is shown that N is unitarily equivalent to the direct sum of the d-tuple of multiplication operators by the coordinate functions acting on the L^2 space with respect to a quasi-invariant measure. This result encompasses a wide range of domains, including irreducible bounded symmetric domains. Motivated by these developments, in the long term, one may also aims to formulate a structure theorem for G-homogeneous subnormal tuples, where G is a locally compact second-countable group, expanding the understanding of such operators in broader settings.