Non-Volterra property of some class of compact operators

Abstract

The authors Matsaev and Mogulskii singled out a wide class of weak perturbation of a positive compact operator $H$, of the form $H(I+S)$, where $S$ is such a compact operator that $I+S$ is continuously invertible, which does not have a nonzero eigenvalue, i.e., is Volterra. On the other hand, such weak perturbations have a complete system of root vectors if the self-adjoint operator $H$ is from the Schatten-von Neumann class. In this paper, we consider the compact operators $A$ representable as a sum of two compact operators $A=C+T$, i.e., $A$ is not necessarily a weak perturbation, where $C$ is a non-negative operator. We will prove existence theorems of nonzero eigenvalues for such operators. The study of the spectral properties of operators generated by differential equations with Cauchy initial data involve, as a rule, Volterra boundary-value problems that are well posed. But Hadamard's example shows that the Cauchy problem for the Laplace equation is ill posed. At present, not a single Volterra well-defined restriction or extension for elliptic-type equations is known. Thus, the following question arises: Does there exist a Volterra well-defined restriction of a maximal operator $\widehat{L}$ or a Volterra well-defined extension of a minimal operator $L_0$ generated by the Laplace operator? The obtained existence theorems for eigenvalues give that a wide class of well-defined restrictions of the maximal operator $\widehat{L}$ and a wide class of well-defined extensions of the minimal operator $L_0$ generated by the Laplace operator cannot be Volterra. Moreover, in the two-dimensional case it is proven that there are no Volterra well-defined restrictions or extensions for Laplace operator at all.