Operators Induced by Radial Measures Acting on the Dirichlet Space

Let D D be the unit disc in the complex plane. Given a positive finite Borel measure μ μ on the radius [0, 1), we let μ ₙ μ n denote the n -th moment of μ μ and we deal with the action on spaces of analytic functions in D D of the operator of Hibert-type H_μ H μ and the operator of Cesàro-type C_μ C μ which are defined as follows: If f is holomorphic in D D , f(z)=∑ ₙ₌₀^∞ aₙzⁿ f ( z ) = ∑ n = 0 ∞ a n z n ( z∈ D) z ∈ D ) , then H_μ (f) H μ ( f ) is formally defined by H_μ (f)(z) = ∑ ₙ₌₀^∞ ( ∑ ₖ₌₀^∞ μ ₙ₊ₖaₖ) zⁿ H μ ( f ) ( z ) = ∑ n = 0 ∞ ∑ k = 0 ∞ μ n + k a k z n ( z∈ D z ∈ D ) and C_μ (f) C μ ( f ) is defined by C_μ (f)(z) = ∑ ₙ₌₀^∞ μ ₙ( ∑ ₖ₌₀ⁿaₖ) zⁿ C μ ( f ) ( z ) = ∑ n = 0 ∞ μ n ∑ k = 0 n