The inverses of solvable extensions and proper restrictions of the operation \(-\text{div\,grad}\) are not Volterra operators (Q5951223)

This paper is devoted to prove that the inverses \(L^{-1}:L^2(Q)\to W^{(2)}_{(2)} (Q)\), where \(Q\) is a smooth bounded domain in \(\mathbb{R}^N\) and \(L\) is some extension of \(-\Delta\), are not Volterra operators (i.e., zero is not the only spectral point). Theorem 1 states that this is the case if \(L\) is a solvable extension of \(-\Delta_{\min}\) and Theorem 2 proves the same result for \(L\) a proper restriction of \(-\Delta_{\max}\). Theorem 3 (which follows from Theorems 1 and 2) gives the same result if \(L\) is a proper operator generated by \(-\Delta\) such that \(L_1\) is bounded.