On Unbounded Integral Operators with Quasisymmetric Kernels

Korotkov, V. B.
Vladikavkaz Mathematical Journal 2020. Vol. 22. Issue 2.

Abstract:
In 1935 von Neumann established that a limit spectrum of self-adjoint Carleman integral operator in \(L_2\) contains \(0\). This result was generalized by the author on nonself-adjoint operators: the limit spectrum of the adjoint of Carleman integral operator contains \(0\). Say that a densely defined in \(L_2\) linear operator \(A\) satisfies the generalized von Neumann condition if \(0\) belongs to the limit spectrum of adjoint operator \(A^{\ast}\). Denote by \(B_0\) the class of all linear operators in \(L_2\) satisfying a generalized von Neumann condition. The author proved that each bounded integral operator, defined on \(L_2\), belongs to \(B_0\). Thus, the question arises: is an analogous assertion true for all unbounded densely defined in \(L_2\) integral operators? In this note, we give a negative answer on this question and we establish a sufficient condition guaranteeing that a densely defined in \(L_2\) unbounded integral operator with quasisymmetric lie in \(B_0\).

Keywords: closable operator, integral operator, kerner of integral operator, limit spectrum, linear integral equation of the first or second kind.

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