Non-Volterra property of a class of compact operators
B. N. Biyarov Institute of Mathematics and Mathematical Modeling, Ministry of Education and Science, Republic of Kazakhstan, Almaty
Abstract:
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.
Keywords:
perturbations, von Neumann–Schatten class, Laplace operator, maximal
(minimal) operator, Volterra operator, well-posed restrictions and
extensions, elliptic operator.
UDC:
517.984
MSC: 47A05,
47A10 Received: 06.09.2023
DOI:
10.4213/mzm14154