Norm-attaining integral operators on analytic function spaces

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Abstract

If f and g are analytic functions in the unit disc D we define Sg(f)(z)=0zf(w)g(w)dw,(zD). If g is bounded then the integral operator Sg is bounded on the Bloch space, on the Dirichlet space, and on BMOA. We show that Sg is norm-attaining on the Bloch space and on BMOA for any bounded analytic function g, but does not attain its norm on the Dirichlet space for non-constant g. Some results are also obtained for Sg on the little Bloch space, and for another integral operator Tg from the Dirichlet space to the Bergman space.

Keywords

Integral operator
Analytic function space
Norm-attaining

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This work was supported by NNSF of China (Grant No. 11171203 & Grant No. 11201280).