Lebesgue and Hardy spaces for symmetric norms I

In this paper, we define and study a class ${\mathcal{R}}_c$ of norms on $L^{\mathrm{\infty }}\left(\mathbb{T}\right)$, called $continuous\ rotationally
symmetric \ norms$,whichproperlycontainstheclass$\left { \left \Vert \cdot \right \Vert _{p}:1\leq p<\infty \right } .$For$\alpha \in $\mathcal{R}$% _{c}$wedefine$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.