Let [phi] : [M.sub.n](S) [right arrow] [M.sub.n](S) be a linear operator. Then [phi] is an invertible linear operator that preserves commuting pairs of matrices if and only if there exist invertible matrix U [member of] [M.sub.n](S) and [f.sub.1], [f.sub.2] [member of] S such that

A calculation of the linear operator occurs in a time domain, by obtaining the time sample convolutions products.

It is also noted that the region of convergence can be increased by a better choice of the auxiliary linear operator.

Also, h [not equal to] 0 an auxiliary parameter and L an auxiliary linear operator. All of [Y.sub.0](t), L and h will be chosen later with great freedom.

The linear operator of degree one [d.sub.A] = d + A : [[OMEGA].sup.i] [right arrow] [Q.sup.i+1] will be called a covariant N-differential induced by a connection form A.

Let 0 [member of] [rho](A), then it is possible to define the fractional power [A.sup.[alpha]], for 0 < [alpha] [less than or equal to] 1, as a closed linear operator on its domain D([A.sup.[alpha]]).

Assume that B can be extended to a bounded linear operator B: X [right arrow] B(X) (denoted by the same symbol B), and that the series [[SIGMA].sup.[infinity].sub.k=1] [[parallel][B.sup.2.sub.k][parallel].sub.B(X)] converge, where [B.sub.k] is the bounded linear operator on X defined by [B.sub.k](x) = (Bx)([e.sub.k]), x [member of] X, k [greater than or equal to] 1.

First of all, let us notice that the extrapolation operator [A.sub.-1] considered in Diagana [1] is bounded from [X.sub.-1] into itself if and only if A is a bounded linear operator on the Banach space X.

Definition 2.2[2]: Let X and Y be two 2-normed spaces and T: X [right arrow] Y be a linear operator. For any e [member of] X, we say that the operator T is e-bounded if there exist [M.sub.e] >0 such that [parallel]T(x), T(e)[parallel] [less than or equal to] [M.sub.e][parallel]x,e[parallel] for all x [member of] X.

Let [sigma](A) denote the spectrum of a matrix A, considered as a bounded linear operator on c.

The method consists of splitting the given equation into linear and non-linear parts, inverting the highest order derivative operator contained in the linear operator on both sides identifying the initial conditions and the terms involving the independent variables alone as initial approximation, decomposing the unknown function into a series whose components can be easily computed, decomposing the non-linear function in terms if polynomial called Adomian's polynomials, and finding the successive terms of the series solution by recurrent relation using the polynomials obtained (cf.

Given a linear operator, T, defined on a dense linear subspace, D(T), of a separable Hilbert space, H, consider the domain of its adjoint: