Some equivalence relations which are Borel reducible to isomorphism between separable Banach spaces

We improve the known results about the complexity of the relation of isomorphism between separable Banach spaces up to Borel reducibility, and we achieve this using the classical spaces $c_0$, ${\mathrm{\ell }}_p$ and $L_p$, $1 \leq p <2$\mathrm{.}Moreprecisely,weshowthattherelation$E_{K_{\sigma}}$ is Borel reducible to isomorphism and complemented biembeddability between subspaces of $c_0$ or ${\mathrm{\ell }}_p,1\le p<2$. We show that the relation $E_{K_{\sigma}} \otimes =^+$ is Borel reducible to isomorphism, complemented biembeddability, and Lipschitz equivalence between subspaces of $L_p,1\le p<2$.