Minimality of eigenfunctions and associated functions of ordinary differential operators

Abstract

Let E, F, H be Banach spaces, $$A,B\in L(E,F)$$ such that $$A+\lambda B$$ and its adjoint $$A^*+\lambda B^*$$ have a biorthogonal system of eigenvectors and associated vectors $$(y_i)_{i=1}^\infty$$ and $$(v_i)_{i=1}^\infty$$ . It is assumed that a closed finite codimensional subspace $$E_0$$ of E exists such that $$E_0$$ is dense in H and such that the restriction of B to $$E_0$$ has a continuous extension to H. It is shown that a subsystem $$(y_i)_{i=k+1}^\infty$$ is minimal in H if $$(v_i)_{i=1}^k$$ has a certain basis property. This abstract result can be applied to regular ordinary linear differential operators in Sobolev spaces whose boundary conditions may depend linearly on the eigenvalue parameter for proving minimality in $$L_p$$ . Explicit examples for Sturm–Liouville problems with one or both boundary conditions depending on $$\lambda$$ are considered.