On quasi-similarity of subnormal operators

For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R2(K,v) denote the closure of Rat(K) in L2(v) and let Sv denote the operator of multiplication by the independent variable z on R2(K, v), that is, Svf = zf for every f∈R2(K, v). SupposeΩis a bounded open subset in the complex plane whose complement has finitely many components and suppose Rat(Ω) is dense in the Hardy space H2(Ω). Letσdenote a harmonic measure forΩ. In this work, we characterize all subnormal operators quasi-similar to Sσ, the operators of the multiplication by z on R2(Ω,σ). We show that for a given v supported onΩ, Sv is quasi-similar to Sσif and only if v/■Ω■σ and log(dv/dσ)∈L1(σ). Our result extends a well-known result of Clary on the unit disk.

On quasi-similarity of subnormal operators

For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R ² (K,v) denote the closure of Rat(K) in L ²(v) and let S _v denote the operator of multiplication by the independent variable z on R ²(K,v), that is, S _v f = zf for every f ∈ R ²(K,v). Suppose Ω is a bounded open subset in the complex plane whose complement has finitely many components and suppose Rat(Ω) is dense in the Hardy space H ²(Ω). Let σ denote a harmonic measure for Ω. In this work, we characterize all subnormal operators quasi-similar to S _σ, the operators of the multiplication by z on R ²(-Ω, σ). We show that for a given v supported on-Ω, S _v is quasi-similar to S _σ if and only if v|∂Ω ≪ σ and log(dv/dσ) ∈ L ¹(σ). Our result extends a well-known result of Clary on the unit disk. This work was supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, the Ministry of Education of China