On unbounded integral operators with quasisymmetric kernels

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 $B_0$.