On unbounded integral operators with quasisymmetric kernels

In 1935 von Neumann established that a limit spectrum of self-adjoint Carleman integral operator in L2 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 L2 linear operator A satisfies the generalized von Neumann condition if 0 belongs to the limit spectrum of adjoint operator A∗. Denote by B0 the class of all linear operators in L2 satisfying a generalized von Neumann condition. The author proved that each bounded integral operator, defined on L2, belongs to B0. Thus, the question arises: is an analogous assertion true for all unbounded densely defined in L2 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 L2 unbounded integral operator with quasisymmetric lie in B0.