Subcouples of codimension one and interpolation of operators that almost agree (Q1881676)

Let \(\bar X\) and \(\bar Y\) be two compatible couples of Banach spaces and let \(T_0: X_0\rightarrow Y_0\) and \(T_1: X_1\rightarrow Y_1\) be bounded linear operators. Let us assume that \(\Gamma\in (\Delta(\overline X))'\) and that \(T_0\) and \(T_1\) agree on \(\Delta(\bar X)\cap \ker \Gamma\), then, in general, we cannot interpolate to obtain that \(T_j\) extends to a bounded operator \(T:J_{\theta, p}(\overline X)\rightarrow J_{\theta, p}(\overline Y)\) where \(J_{\theta, p}\) is the classical \(J\)-method of interpolation. However, in this paper, the author defines a space \(J_{\theta, p; \Gamma}(\overline X)\) in such a way that \(T:J_{\theta, p; \Gamma}(\overline X)\rightarrow J_{\theta, p}(\overline Y)\) is bounded and he studies this new space determining, among other things, when \(J_{\theta, p; \Gamma}(\overline X)= J_{\theta, p}(\overline X)\) or \(J_{\theta, p; \Gamma}(\overline X)\) is a closed subspace of \(J_{\theta, p}(\overline X)\). He also defines the space \(J_{E; \Gamma}(\overline X)\) for the general real method of interpolation.