  A method is suggested for the calculation of the DeWitt-Seeley-Gilkey (DWSG)
coefficients for the operator $\sqrt{-\nabla^2 + V(x)}$ basing on a
generalization of the pseudodifferential operator technique. The lowest DWSG
coefficients for the operator $\sqrt{-\nabla^2} + V(x)$ are calculated by using
the method proposed. It is shown that the method admits a generalization to the
case of operators of the type $(-\nabla^2 + V(X))^{1/{\rm m}}$, where m is an
arbitrary rational number. A more simple method is proposed for the calculation
of the DWSG coefficients for the case of strictly positive operators under the
sign of root. By using this method, it is shown that the problem of the
calculation of the DWSG coefficients for such operators is exactly solvable.
Namely, an explicit formula expressing the DWSG coefficients for operators with
root through the DWSG coefficients for operators without root is deduced.
