Lebesgue and Hardy spaces for symmetric norms I

Abstract

In this paper, we define and study a class ${\mathcal{R}}_c$ of norms on $L^{\infty }\left(\mathbb{T}\right)$, called $continuousrotationallysymmetricnorms$, which properly contains the class $\left\{{\Vert \cdot \Vert }_p:1\le p<\infty \right\}.$ For $\alpha \in \mathcal{R}$ we define $L^{\alpha }\left(\mathbb{T}\right)$ and the Hardy space $H^{\alpha }\left(\mathbb{T}\right)$, and we extend many of the classical results, including the dominated convergence theorem, convolution theorems, dual spaces, Beurling-type invariant spaces, inner-outer factorizations, characterizing the multipliers and the closed densely-defined operators commuting with multiplication by $z$. We also prove a duality theorem for a version of $L^{\alpha }$ in the setting of von Neumann algebras.

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