Cesàro-like operators acting on spaces of analytic functions

Let D D be the unit disc in C C . If μ μ is a finite positive Borel measure on the interval [0, 1) and f is an analytic function in D D , f(z)=∑ ₙ₌₀^∞ aₙzⁿ f ( z ) = ∑ n = 0 ∞ a n z n ( z∈ D z ∈ D ), we define aligned C_μ (f)(z)= ∑ ₙ₌₀^∞ μ ₙ( ∑ ₖ₌₀ⁿaₖ) zⁿ, z∈ D, aligned C μ ( f ) ( z ) = ∑ n = 0 ∞ μ n ∑ k = 0 n a k z n , z ∈ D , where, for n≥ 0 n ≥ 0 , μ ₙ μ n denotes the n -th moment of the measure μ μ , that is, μ ₙ=∫ [0, 1)tⁿdμ (t). μ</mml:m