On an embedding criterion for interpolation spaces and application to indefinite spectral problems

An embedding criterion for interpolation spaces is formulated and applied to the study of the Riesz basis property in the $L_{2,|g|}$ space of eigenfunctions of an indefinite Sturm–Liouville problem $u''=\lambda gu$ on the interval $(-1,1)$ with the Dirichlet boundary conditions, provided that the function $g(x)$ changes sign at the origin. In particular, the basis property criterion is established for an odd $g(x)$. Some connections with stability in interpolation scales are discussed.