Non-well posed evolution equations and Fredholm operators
We shall consider the operator (Fξ)(t) = ξ(t) + A(t)ξ(t) (*) where A(t) ∈ L(W,H) is continuously differentiable in the uniform operator topology with W H, a continuous dense injection. Both W and H are Hilbert spaces each with its own norm. At the same time we assume that W is a dense linear subspace of H, with A(t) selfadjoint when regarded as an unbounded operator on H and domain D(A(t)) = W. We consider F as an operator F:W1,2(R;H)∩ L2(R;W) → L2(R;W) (**) and show that this is Fredholm provided W ⊂ H is a compact embedding and the limit operator A±limt→±∞ = A(t) is bijective.
Journal of the Nigerian Association of Mathematical Physics Vol. 9 2005: pp. 43-48