Non-Volterra Property of a Class of Compact Operators

The authors Matsaev and Mogulskii identified a wide class of weak perturbations of a positive compact operator $$H$$ that have no nonzero eigenvalues, i.e., are Volterra operators. By a weak perturbation of a positive operator $$H$$ we mean an operator of the form $$H(I+S)$$ , where $$S$$ is a compact operator such that $$I+S$$ is continuously invertible. On the other hand, these weak perturbations have a complete system of root vectors if the self-adjoint operator $$H$$ belongs to a von Neumann–Schatten class. In this paper, we consider compact operators $$A$$ that can be represented as the sum of two compact operators $$A=C+T$$ (i.e., $$A$$ is not necessarily a weak perturbation), where $$C$$ is a positive operator. In this paper, we prove theorems on the existence of nonzero eigenvalues for such operators. As is known, Cauchy problems for differential equations are, as a rule, well-posed Volterra problems. However, Hadamard’s example shows that the Cauchy problem for the Laplace equation is ill posed. Up to now, not a single Volterra well-posed restriction or extension is known for an elliptic-type equation. Thus, the following question arises: “Does there exist a Volterra well-posed restriction of the maximal operator $$\widehat{L}$$ or a Volterra well-posed extension of the minimal operator $$L_0$$ generated by elliptic-type equations?” The abstract theorems on the existence of eigenvalues obtained here show that a wide class of well-posed restrictions of the maximal operator $$\widehat{L}$$ and a wide class of well-posed extensions of the minimal operator $$L_0$$ generated by elliptic-type equations cannot be Volterra operators. Moreover, in the two-dimensional case, it is proved that, for the Laplace operator, there are no well-posed Volterra restrictions and extensions at all.