Multiplication operators on Hardy and weighted Bergman spaces over planar regions

Abstract

This paper studies some aspects of commutant theory and functional calculus for analytic multiplication operators on Hardy and weighted Bergman spaces over bounded planar regions. Multiplication operators defined by univalent functions are shown to commute only with multiplication operators. This result is generalized to a tuple of operators, and a sufficient condition is given for irreducibility of that induced by finite Blaschke products. Operators defined by fairly general ancestral functions are shown to commute with no nonzero compact operators, and these include the ones by monomial functions over annuli. For such operators over annuli, we characterize a certain dense subalgebra of the commutant. Norm and sequential weak closures of the analytic functional calculus algebra generated by a multiplication operator are characterized and essential spectral mapping properties obtained. Generalizing the similarity for finite Blaschke products to a larger class of weighted Bergman spaces, the commutant classification of these operators is obtained and seen strictly finer than the similarity classification, among other related results.