(1) Let B : U [right arrow] X be a bounded linear operator
For a bounded linear operator
A [member of] L(X), let [rho](A) and D(A) stand for the resolvent and domain of A, respectively.
B : [H.sub.1] [right arrow] [H.sub.2] is a bounded linear operator
with adjoint [B.sup.*].
The linear space of all bounded linear operators
from X into Y is denoted by L(X, Y) and denote L(X, X) by L(X).
[T.sub.n] and T are bounded linear operators
from X into X.
In this paper we study problem (1) for the case where the bounded linear operator
A is singular but g-Drazin invertible.
[greater than or equal to] [[chi].sub.n], and [r.sub.+][P.sub.+][r.sub._]I ([r.sub._][P.sub.+][r.sub.+]I) represents a bounded linear operator
in [[[L.sub.2](T)].sub.n]; the number [chi] = [n.[summation] over j=1] [[chi].sub.j]is called the factorization index of the determinant of the matrix function r.
Let A : [H.sub.1] [right arrow] [H.sub.2] be a bounded linear operator
. Assume that [F.sub.1] : C x C [right arrow] R and [F.sub.2] : Q x Q [right arrow] R are the bifunctions satisfying Assumption 7 and F2 is upper semicontinuous in the first argument.
Let X be any bounded linear operator
mapping [H.sup.2](d[micro]) into itself.
Let H and K be Krein spaces and let T : H [right arrow] K be a bounded linear operator
. We say that T is a contraction if for all x [member of] H, [<Tx, Tx>.sub.K] [less than or equal to] [<x, x>.sub.H].
A bounded linear operator
A is said to be quasi-Hermitian if its imaginary component
Then f is said to be [[omega].sup.*]-Gateaux differentiable at x [member of] N(C) if there exists a bounded linear operator
[T.sub.x] : C [right arrow] [Y.sup.*] such that for every h [member of] [C.sub.x] and every y [member of] Y the following limit exists: