Let H be a real or complex Hilbert space. The set  consisting of positive bounded isomorphism, is an open subset of  the Banach space consisting of self-adjoint bounded operators. For any  there exists an  such that  R is called the positive square root of L. This Rcan be expressed as

where G is a path containing  in its interior and g is a convenient holomorphic function. Denote by  the positive square root of L. The main goal of this work is to prove that  defined by  is a -diffeomorphism, and moreover, we shall show that for any  the derivative    is defined by

spectraltheory,square root of nonnegative operators, complexification of operators, Cauchy’s integral formula.