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We study norm estimates of matrix operators between Banach spaces. In particular, we prove a general variant of the HardyLittlewood result on lower estimate for norms of operators between p -spaces and show applications to Calderón-Lozanovskii sequence lattices. We also provide estimates for norms of radially generated operators from abstract Hardy spaces to Banach sequence lattices. Likewise we prove the mean convergence theorem in abstract Hardy spaces and show general examples of radially generated operators.

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Abstract. The spectra of invertible weighted composition operators uCϕ on the Bloch and Dirichlet spaces are studied. In the Bloch case we obtain a complete description of the spectrum when ϕ is a parabolic or elliptic automorphism of the unit disc. In the case of a hyperbolic automorphism ϕ, exact expressions for the spectral radii of invertible weighted composition operators acting on the Bloch and Dirichlet spaces are derived.