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Norm-attaining integral operators on analytic function spaces. (English) Zbl 1257.30061

Summary: If f and g are analytic functions in the unit disc 𝔻, we define

S g (f)(z)= 0 z f ' (w)g(w)dw,z𝔻·

If g is bounded, then the integral operator S g is bounded on the Bloch space, on the Dirichlet space, and on BMOA. We show that S g 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 S g on the little Bloch space, and for another integral operator T g from the Dirichlet space to the Bergman space.

30H20Bergman spaces, Fock spaces
30H30Bloch spaces
47A30Operator norms and inequalities
47A10Spectrum and resolvent of linear operators