For analytic functions g on the unit disk with non-negative Maclaurin coefficients, we describe the boundedness and compactness of the integral operator $$T_g(f)(z)=\int _0^zf(\zeta )g'(\zeta )\,d\zeta $$ T g ( f ) ( z ) = ∫ 0 z f ( ζ ) g ′ ( ζ ) d ζ from a space X of analytic functions in the unit disk to $$H^\infty $$ H ∞ , in terms of neat and useful conditions on the Maclaurin coefficients of g. The choices of X that will be considered contain the Hardy and the Hardy–Littlewood spaces, the Dirichlet-type spaces $$D^p_{p-1}$$ D p - 1 p , as well as the classical Bloch and $$\mathord {\mathrm{BMOA}}$$ BMOA spaces.

Bounded and compact differences of two composition operators acting from the weighted Bergman space $$A^p_\omega $$ A ω p to the Lebesgue space $$L^q_\nu $$ L ν q , where $$0<q<p<\infty $$ 0 < q < p < ∞ and $$\omega $$ ω belongs to the class "Equation missing" of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proofs a new description of q-Carleson measures for $$A^p_\omega $$ A ω p , with $$p>q$$ p > q and "Equation missing", involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of q-Carleson measures for the classical weighted Bergman space $$A^p_\alpha $$ A α p with $$-1<\alpha <\infty $$ - 1 < α < ∞ to the setting of doubling weights. The case "Equation missing" is also briefly discussed and an open problem concerning this case is posed.

For analytic functions g on the unit disc with non-negative Maclaurin coefficients, we describe the boundedness and compactness of the integral operator Tgpf qpzq " ş z 0 f pζqg 1 pζq dζ from a space X of analytic functions in the unit disc to H 8 , in terms of neat and useful conditions on the Maclaurin coefficients of g. The choices of X that will be considered contain the Hardy and the Hardy-Littlewood spaces, the Dirichlet-type spaces D p p´1 , as well as the classical Bloch and BMOA spaces.Hpβq g,z in the dual space of X is often laborious if not even frustrating. Because of these reasons, in this study we restrict ourselves to the case in which the symbol g has non-negative Maclaurin coefficients, and search for neat and useful conditions in terms of the Maclaurin coefficients of g that can be used to test if T g is either bounded or compact from X to H 8 . The starting point is the characterization [4, Theorem 2.2] given above, and the choices for X that will be considered in the sequel contain the Hardy and the Hardy-Littlewood spaces, and certain Dirichlet-type spaces, as well as the