**Commutants and Reflexivity of Multiplication tuples on Vector-valued Reproducing Kernel Hilbert Spaces**

In this talk, we address the question of identifying commutant and reflexivity of the multiplication d-tuple

M_{z} on a reproducing kernel Hilbert space H of E-valued holomorphic functions on Ω, where E is a separable Hilbert space and Ω is a bounded domain in C^{d} admitting bounded approximation by polynomials. In case E is a finite dimensional cyclic subspace for M_{z}, under some natural conditions on the B(E)-valued kernel associated with H, the commutant of M_{z} is shown to be the algebra H^{∞}_{B(E)}(Ω) of bounded holomorphic B(E)-valued functions on Ω, provided M_{z} satisfies the matrix-valued von Neumann’s inequality. Also, we show that a multiplication d-tuple M_{z} on H satisfying the von Neumann’s inequality is reflexive.

The talk is based on joint work with Sameer Chavan and Shailesh Trivedi.