Invertibility of operators in spaces of real interpolation
The main result of the paper is the following Theorem. Let \(A\) be a bounded linear operator from a Banach couple \((X_0, X_1)\) to a Banach couple \((Y_0, Y_1)\) such that \(A\) is invertible on the spaces \(X_0\) and \(X_1\). Suppose also that its kernel \(\ker A\subset X_0+X_1\) is finite-dimensional and has a basis \(e_1,\dots, e_n\) such that \(K(t,e_i; X_0, X_1)\approx t^{\theta_i}\) \((\theta_i\in (0,1)\), \(\theta_i\neq \theta_j\) for \(i\neq j\). Then the operator \(A\) is invertible on the space \((X_0, X_1)_{\theta,q}\) if and only if \(\theta\neq \theta_i\) \((i=1,\dots,n).\) Here, \(K(t,f;X_0,X_1)= \inf_{f=f_0+f_1}(\| f_0\| _{X_0} +t \| f_1\| _{X_1})\) and \((X_0, X_1)_{\theta,q}=\{f\in X_0+X_1: (\int_0^\infty [t_{-\theta} K(t,f;X_0,X_1)]^q \,dt/t)^{1/q}<\infty\}.\)
scientific article; zbMATH DE number 5289686