Ordinary differential operators under Stieltjes boundary conditions

Abstract:

The operator ${L_p}y = y’ + Py$, whose domain is determined in part by the Stieltjes integral boundary condition $\int _0^1 {d\nu (t)y(t) = 0}$, is studied in $\mathcal {L}_n^p(0,1),1 \leqslant p < \infty$. It is shown that ${L_p}$ has a dense domain; hence there exists a dual operator $L_q^ +$ operating on $\mathcal {L}_n^q(0,1)$. After finding $L_q^ +$ we show that both ${L_p}$ and $L_q^ +$ are Fredholm operators. This implies some elementary results concerning the spectrum and states of ${L_p}$. Finally two eigenfunction expansions are derived.