Interpolation of Closed Ideals of Bilinear Operators

Abstract

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