Commutants and reflexivity of multiplication tuples on vector-valued reproducing kernel Hilbert spaces

摘要

Motivated by the theory of weighted shifts on directed trees and its multivariable counterpart, we address the question of identifying commutant and reflexivity of the multiplicationd-tupleMzon a reproducing kernel Hilbert spaceHofE-valued holomorphic functions on Ω, whereEis a separable Hilbert space and Ω is a bounded domain inCdadmitting bounded approximation by polynomials. In caseEis a finite dimensional cyclic subspace forMz, under some natural conditions on theB(E)-valued kernel associated withH, the commutant ofMzis shown to be the algebraHB(E)∞(Ω)of bounded holomorphicB(E)-valued functions on Ω, providedMzsatisfies the matrix-valued von Neumann's inequality. This generalizes a classical result of Shields and Wallen (the case ofdim?E=1andd=1). As an application, we determine the commutant of a Bergman shift on a leafless, locally finite, rooted directed treeTof finite branching index. As the second main result of this paper, we show that a multiplicationd-tupleMzonHsatisfying the von Neumann's inequality is reflexive. This provides several new classes of examples as well as recovers special cases of various known results in one and several variables. We also exhibit a family of tri-diagonalB(C2)-valued kernels for which the associated multiplication operatorsMzare non-hyponormal reflexive operators with commutants equal toHB(C2)∞(D).