Eigenfunction expansions for a class of J-selfadjoint ordinary differential operators with boundary conditions containing the eigenvalue parameter

SynopsisIn provided with a J-innerproduct we characterize the J-selfadjoint operators generated by a symmetric ordinary differential expression on an open real interval ι. For a subclass of these operators we prove eigenfunction expansion results using Hilbertspace-techniques.</jats:p

Adjoint Subspaces in Banach Spaces, with Applications to Ordinary Differential Subspaces

Given two subspaces A0 ⊂ A1 ⊂ W = X ⊕ Y, where X, Y are Banach spaces, we show how to characterize, in terms of generalized boundary conditions, those adjoint pairs A, A* satisfying A0 ⊂ A ⊂ A1, A1* ⊂ A* ⊂ A0* ⊂ W+ = Y* ⊕ X*, where X*, Y* are the conjugate spaces of X, Y, respectively. The characterizations of selfadjoint (normal) subspace extensions of symmetric (formally normal) subspaces appear as special cases when Y = X*. These results are then applied to ordinary differential subspaces in W = Lq(ι) ⊕ Lr(ι), 1 ≤ q, r ≤ ∞, where ι is a real interval, and in W = C(ī) ⊕ C(ī), where ī is a compact interval.