Interpolation of closed ideals of bilinear operators

Abstract

We extend the (outer) measure $\gamma_{_{\mathcal{I}}}$ associated to an operator ideal $\mathcal{I}$ to a measure $\gamma_{_{\mathfrak{I}}}$ for bounded bilinear operators. If $\mathcal{I}$ is surjective and closed, and $\mathfrak{I}$ is the class of those bilinear operators such that $\gamma_{_{\mathfrak{I}}}(T)=0$, we prove that $\mathfrak{I}$ coincides with the composition bideal $\mathcal{I}\circ \mathfrak{B}$. If $\mathcal{I}$ satisfies the $\Sigma_r$-condition, we establish a simple necessary and sufficient condition for an interpolated operator by the real method to belong to $\mathfrak{I}$. Furthermore, if in addition $\mathcal{I}$ is symmetric, we prove a formula for the measure $\gamma_{_{\mathfrak{I}}}$ of an operator interpolated by the real method. In particular, results apply to weakly compact operators.