Interpolation of Closed Subspaces and Invertibility of Operators

Abstract

Let $(Y_0,Y_1)$ be a Banach couple and let $X_j$ be a closed complemented subspace of $Y_j$, ($j=0,1$). We present several results for the general problem of finding necessary and sufficient conditions on the parameters $(\theta ,q)$ such that the real interpolation space $(X_0,X_1{)}_{\theta ,q}$ is a closed subspace of $(Y_0,Y_1{)}_{\theta ,q}.$ In particular, we establish conditions which are necessary and sufficient for the equality $(X_0,X_1{)}_{\theta ,q}=(Y_0,Y_1{)}_{\theta ,q},$ with the proof based on a previous result by Asekritova and Kruglyak on invertibility of operators. We also generalize the theorem by Ivanov and Kalton where this problem was solved under several rather restrictive conditions, such as that $X_1=Y_1$ and $X_0$ is a subspace of codimension one in $Y_0$.