Nagumo [6] developed a degree theory in a setting of linear convex topological space Y for operators of the type I - B, where B : [bar.G] [right arrow] Y is a
compact operator, I is the identity mapping on Y, and G is a nonempty and open subset of X.
Let K : [A.sup.p.sub.[alpha]] [right arrow] [B.sub.[mu]] be any
compact operator. Then
Let S : [L.sup.[infinity]] (0, b) [right arrow] C[0, b] be a linear
compact operator. Let [P.sub.N]: C[0,b] [right arrow] [S.sup.(-1).sub.m-1] ([[PI].sub.N]) (N [member of] N) be defined by (5.1).
The perturbation of a polaroid operator by a
compact operator may or may not effect the polaroid property of the operator.
If (C(t))t[greater than or equal to] 0 is a compact cosine family, then P is a continuous and
compact operator.
In this context, recall that for a
compact operator Q [member of] L(H), Fredholm's theorem states that for all [lambda] [member of] R, ([lambda]I - Q)x = y is solvable for y [member of] H if and only if y [perpendicular to] N([lambda]I - Q*), i.e., if ([lambda], v) is an eigenpair of Q* then it must hold that [??]y, v[??] = 0.
We will prove that F is a
compact operator. According to (4.12), F([OMEGA]) | [0T] is relatively compact in C([0,T], E).
(ii) Assume that the sequence {[T.sub.n]} of bounded operators satisfies that there is at least one
compact operator within all subsequences [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] being subject to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some subsequence {[j.sub.n]} [subset] [N.sub.0] for any n [member of] [N.sub.0].
Here we prove a lemma which is similar to Riesz lemma in normed space and (e) using this we list the properties of
compact operator.
Therefore we have to transfer S into another continuous and
compact operator T.