Non-Volterra Property of a Class of Compact Operators
Biyarov B.N.
December 2024Pleiades Publishing
Mathematical Notes
2024#116Issue 5920 - 933 pp.
Abstract: The authors Matsaev and Mogulskii identified a wide class of weak perturbations of a positive compact operator that have no nonzero eigenvalues, i.e., are Volterra operators. By a weak perturbation of a positive operator we mean an operator of the form, where is a compact operator such that is continuously invertible. On the other hand, these weak perturbations have a complete system of root vectors if the self-adjoint operator belongs to a von Neumann–Schatten class. In this paper, we consider compact operators that can be represented as the sum of two compact operators (i.e., is not necessarily a weak perturbation), where 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 or a Volterra well-posed extension of the minimal operator 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 and a wide class of well-posed extensions of the minimal operator 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.
elliptic operator , Laplace operator , maximal (minimal) operator , perturbations , Volterra operator , von Neumann–Schatten class , well-posed restrictions and extensions
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Institute of Mathematics and Mathematical Modeling, Ministry of Education and Science, Republic of Kazakhstan, Almaty, 050010, Kazakhstan
Institute of Mathematics and Mathematical Modeling
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