Given a sequence of orthonormal polynomials \(\{P_n\}\), satisfying the recurrence relation \[ a_nP_{n+1}(x)+b_nP_n(x)+a_nP_{n-1}(x)=xP_n(x), \quad a_n>0, \quad b_n \in {\mathbb{R}}, \] with \(P_0=0\), \(P_1=1\), the corresponding Jacobi matrix \(T\) is defined by \[ Te_1=a_1e_{2}+b_1e_1, \qquad Te_n=a_ne_{n+1}+b_ne_n+a_ne_{n-1}, \quad n>1, \] where \(e_n, n \geq 1\), is the standard orthonormal basis in \(l_2\). It is well known that the uniqueness of the measure of orthogonality for \(\{ P_n \}\) is equivalent to the essential self-adjointness of \(T\). The approach followed in this paper is based on the separation \(T=AV^*+VA+B\), where \(A\) and \(B\) are diagonal and \(V\) is the right shift operator. The author studies in detail the definition domains of \(A\), \(B\), \(VA\) and \(AV^*\). In this way, a new sufficient condition for non-selfadjointness of \(T\) is proved, which does not restrict the spectrum of \(T\) to the semiaxis.