The author considers an operator \(A\) in the infinite tensor product of \(L_2\)-spaces which has the form of a separating tensor product of a sequence of self-adjoint operators \(A_k\) (that is an infinite sum of terms \(I\otimes I\otimes \ldots \otimes I\otimes A_k\otimes I\otimes \ldots\), with \(A_k\) at the \(k\)-th place). It is shown that the Kato inequality for each operator acting on a finite number of variables (or, under some additional assumptions, for each operator \(A_k\)) implies the Kato inequality for the operator \(A\). This result is then used in order to obtain a sufficient condition of the essential self-adjointness for perturbations of the operator \(A\).