Resumen de Ultrasymmetric sequence spaces in approximation theory
Evgeniy Pustylnik
Let $\varphi(t)$ be a positive increasing function and let $\widetilde E$ be an arbitrary sequence space, rearrangemen-tinvariant with respect to the atomic measure $\mu(n) = 1/n$. Let $\{a^\ast_n\}$ mean the decreasing rearrangement of a sequence $\{\mid a_n|\}$. A sequence space $\ell_{\varphi,E}$ with symmetric (quasi)norm $\parallel\{\varphi(n)a^\ast_n\}\parallel_ {\widetilde E}$ is called ultrasymmetric, because it is not only intermediate but also interpolation between the corresponding Lorentz and Marcinkiewicz spaces $\Lambda_\varphi$ and $M_\varphi$. We study properties of the spaces $\ell_{\varphi,E}$ for all admissible parameters $\varphi,E$ and use them for the definition of ultrasymmetric approximation spaces $X_{\varphi,E}$, which essentially generalize most of classical approximation spaces. At the same time we show that the spaces $X_{\varphi,E}$ possess almost all properties of classical prototypes, such as equivalent norms, representation, reiteration, embeddings, transformation etc. Special attention is paid to interpolation properties of these spaces. At last, we apply our results to ultrasymmetric operator ideals.
