Interpolation of the inner measure of bilinear operators by the real method

Abstract

We describe a procedure for extending the inner measure $\beta_{_{\mathcal{I}}}$ associated to an operator ideal $\mathcal{I}$ to a measure $\beta_{_{\mathfrak{J}}}$ for bounded bilinear operators $T$. When $\mathcal{I}$ is injective and close, we show that $\beta_{_{\mathfrak{J}}}(T)=0$ if and only if $T=RS$ for some bounded bilinear operator $S$ and $R\in\mathcal{I}$. If $\mathcal{I}$ satisfies the $\Sigma_r$-condition, then we establish a convexity inequality for the measure $\beta_{_{\mathfrak{J}}}$ of a bilinear operator interpolated by the real method.