We present factorizations of weighted Lebesgue, Cesàro and Copson spaces, for weights satisfying the conditions which assure the boundedness of the Hardy's integral operator between weighted Lebesgue spaces. Our results enhance, among other, the best known forms of weighted Hardy inequalities.

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We find the best constants in inequalities relating the standard norm, the dual norm, and the norm∥x∥(p,s):=inf⁡{∑k∥x(k)∥p,s}, where the infimum is taken over all finite representationsx=∑kx(k)in the classical Lorentz sequence spaces. A crucial point in this analysis is the concept of level sequence, which we introduce and discuss. As an application, we derive the best constant in the triangle inequality for such spaces.

We introduce the Besov-Schatten spaces B p 2 , a matrix version af analytic Besov space, and we compute the dual of this space showing that it coincides with the matricial Bloch space introduced previously in Popa 2007. Finally we compute the space of all Schur multipliers on B 1 2 .

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