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Carleson measures and a class of generalized integration operators on the Bergman space. (English) Zbl 1225.47032

Summary: We consider a linear operator

I h,ϕ (n) f(z)= 0 z f (n) (ϕ(ξ))h(ζ)dζ

induced by holomorphic maps h and ϕ of the open unit disk 𝔻, where ϕ(𝔻)𝔻 and n is a non-negative integer. A complete characterization of when I h,ϕ (n) is bounded on the Bergman space 𝒜 2 is established by using Luecking’s result for Carleson measures. We also compute upper and lower bounds for the essential norm of this operator on the Bergman space.

MSC:
47B35Toeplitz operators, Hankel operators, Wiener-Hopf operators
46E20Hilbert spaces of continuous, differentiable or analytic functions
30H20Bergman spaces, Fock spaces