Reducing subspaces of multiplication operators on the Dirichlet space

2018/06/28 by Shuaibing Luo, Luo, Shuaibing
Mathematics · #Advanced Harmonic Analysis Research #FOS: Mathematics #Finite Group Theory Research #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1806.10753

openalex publication_date 2018/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the reducing subspaces for the multiplication operator by a finite Blaschke product ϕ on the Dirichlet space D. We prove that any two distinct nontrivial minimal reducing subspaces of M_ϕ are orthogonal. When the order n of ϕ is 2 or 3, we show that M_ϕ is reducible on D if and only if ϕ is equivalent to zⁿ. When the order of ϕ is 4, we determine the reducing subspaces for M_ϕ, and we see that in this case M_ϕ can be reducible on D when ϕ is not equivalent to z⁴. The same phenomenon happens when the order n of ϕ is not a prime number. Furthermore, we show that M_ϕ is unitarily equivalent to M_zⁿ (n > 1) on D if and only if ϕ= azⁿ for some unimodular constant a.

Related

• Reducing subspaces for multiplication operators on the Dirichlet space through local inverses and Riemann surface
• Reducing subspaces for analytic multipliers of the Bergman space
• Multiplication operators on the Bergman space via analytic continuation
• Multiplication by a finite Blaschke product on weighted Bergman spaces: commutant and reducing subspaces
• A Z^k-invariant subspace without the wandering property