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
Mz 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 Cd admitting bounded approximation by polynomials. In case E is a finite dimensional cyclic subspace for Mz, under some natural conditions on the B(E)-valued kernel associated with H, the commutant of Mz is shown to be the algebra H∞B(E)(Ω) of bounded holomorphic B(E)-valued functions on Ω, provided Mz satisfies the matrix-valued von Neumann’s inequality. Also, we show that a multiplication d-tuple Mz on H satisfying the von Neumann’s inequality is reflexive.
The talk is based on joint work with Sameer Chavan and Shailesh Trivedi.