Reducing subspaces for analytic multipliers of the Bergman space

2011/10/21 by Ronald G. Douglas, Douglas, Ronald G., Mihai Putinar +3
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #math.CV #math.FA

paper · pdf · doi:10.48550/arxiv.1110.4920

arxiv created 2011/10/21 · arxiv updated 2011/10/25

Abstract

We answer affirmatively the problem left open in \citeDSZ,GSZZ and prove that for a finite Blaschke product ϕ, the minimal reducing subspaces of the Bergman space multiplier M_ϕ are pairwise orthogonal and their number is equal to the number q of connected components of the Riemann surface of ϕ^-1∘ ϕ. In particular, the double commutant \M_ϕ,M_ϕ^∗\' is abelian of dimension q. An analytic/arithmetic description of the minimal reducing subspaces of M_ϕ is also provided, along with a list of all possible cases in degree of ϕ equal to eight.

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