SQUARE POWERS OF SINGULARLY PERTURBED OPERATORS

We use the method of self-adjoint extensions to define a self-adjoint operator A(T) as the singular perturbation of a given self-adjoint ope rator A by a singular operator T on a Hilbert space. We also find the structure of a singular operator e such that the singular perturbation of A(2) by Q satisfies (A(2))(Q) = (A(r))(2) We obtain the explicit f orm of Q in terms of A and T. A definition of the n-th power for stric tly positive symmetric operators is also given.