Square Powers of Singularly Perturbed Operators

Abstract

Abstract 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 operator A by a singular operator T on a Hilbert space. We also find the structure of a singular operator Q such that the singular perturbation of A 2 by Q satisfies ( A 2 ) Q = ( A T ) 2 . We obtain the explicit form of Q in terms of A and T . A definition of the n ‐th power for strictly positive symmetric operators is also given.