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 selfadjoint operator $H$ belongs to the 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.