Recall that abounded linear operator A on a Hilbert space is a

Fredholm operator if and only if ran A is closed and both ker A and ker [A.sup.*] are finite dimensional.

Since [mu]I - [[??].sub.[phi],b] ia a

Fredholm operator of index 0 and [mu] does not belong to the spectrum, the claim follows from Fredholm Alternative.

Finally, we remark that the

Fredholm operator, given by

Suppose that L : D(L) [subset] X [right arrow] X is a

Fredholm operator with index zero and N : [bar.[OMEGA]] [right arrow] X is L-compact on [bar.[OMEGA]], where [OMEGA] is an open bounded subset of X.

Let L be a

Fredholm operator of index zero and let N be L-compact on [bar.[OMEGA]].

Therefore, system (8) is equivalent to the following

Fredholm operator equation of the second kind:

An operator T is called a

Fredholm operator if the range of T denoted by ran(T) is closed and both ker T and ker [T.sup.*] are finite dimensional and is denoted by T [member of] [PHI](H).

Recall that to a

Fredholm operator T : X [right arrow] Y between Banach spaces is associated a unique number, called the index, defined by the formula ind(T) = dim ker(T) - dim(Y/X).

Let A : Dom A [subset] X [right arrow] X be a

Fredholm operator of index zero, [OMEGA] a bounded open subset of a Banach space X and let B(I - Q)F be a compact operator from [bar.[OMEGA]] to X.

By Lemma 2.1, L is a

Fredholm operator of index zero and N is L-compact on [bar.[OMEGA]].

Since it is known that [[sigma].sub.e]([B.sub.z]) = T, for any [lambda] [member of] D, it follows that [lambda]I-[B.sub.z] is

Fredholm operator. Let [??] be an operator defined on u[H.sup.2] [direct sum] [[bar.H].sup.2.sub.0] as follows

Index of a

Fredholm operator [mathematical expression not reproducible] is equal to 0.