where [mathematical expression not reproducible] denotes the orthogonal projection in [L.sup.2]([OMEGA]) onto the admissible set of the control and [B.sup.*] is the

adjoint operator of B.

The matrix-vector multiplication should be replaced with the specifically designed function to perform the equivalent of the forward operator [g.sub.A] and

adjoint operator [g.sup.H.sub.A].

The main properties of the

adjoint operator were stated in Ch.

A : [H.sub.1] [right arrow] [H.sub.3] is a bounded linear operator with its

adjoint operator [A.sup.*], [sigma] [member of] (0,1) is a parameter controlling step length, 0 < [gamma] < [eta] < 1/(1 + [square root of (1 + [L.sup.2])]), and {[[alpha].sub.n]} [subset] (0,1).

Throughout the paper, we will assume that A(t) is invertible, A(t) = [A.sup.*](t) for all [mathematical expression not reproducible] for all t [member of] [q.sup.N], where * denotes the

adjoint operator. Note that we can also define the operator L using the infinite matrix

We use the same notation H to denote this self

adjoint operator. The operator admits the partial wave expansion.

PA = A[P.sup.*] where [P.sup.*] is the

adjoint operator of P with respect to [<x, x>.sub.H], defined by [P.sup.*]x = x - U[<U, AU>.sup.-1.sub.H][<AU,x>.sub.H].

(i) [??] (ii) Let X and Y be left Banach G-modules and suppose that T : X [right arrow] Y is a bounded linear G-module map and [T.sup.*] : [Y.sup.*] [right arrow] [X.sup.*] is its

adjoint operator. For [mu] [member of] M(G), x [member of] X and [phi] [member of] [Y.sup.*],

It is claimed that the mapping f is contraction with constant [beta] = 1/2, A is a bounded linear operator on R with

adjoint operator [A.sup.*] and [parallel]A[parallel] = [parallel][A.sup.*] [parallel] = 1/2, and B is a strongly positive bounded linear self-adjoint operator with constant [bar.[gamma]] = 1 on R.

In the theory of inverse problems, we are asked to find the solution of the operator equation Bf = g where B: H [right arrow] H is a self

adjoint operator on a Hilbert space H, and g e H is the exact datum.

Recall that the

adjoint operator [T.sup.*] is then defined by [T.sup.*][phi] = [[phi].sup.*] for all [phi] [member of] D([T.sup.*]).