162 D. SALAMON, FLOER HOMOLOGYdenotes the Laplace operator. Once this is established, the proof of Lemma 2.3
is an easy exercise. Details are left to the reader. Hint: Use the formula (8s -
Jo8t)(8s + J 08 t) = b.. Consult Appendix Bin [31] if you get stuck. 0
Lemma 2.4. Suppose that S(s, t) = S(t) is independent of s and that
det(n - w(l)) # o
where w: [O, 1] ___, Sp(2n) is defined by ~(t) = JoS(t)w(t) and '11(0) = Il. Then the
operatorD = 8s + Jo8t + S: W^1 ·P(IR x S1,IR^2 n) ___, £P(IR x S^1 ,IR^2 n)
is bijective for 1 < p < oo.
Proof. We shall only consider the case p ?: 2. The proof consists of four steps.
Step 1: The result holds for p = 2.
Consider the operatorA= Jo8t + S: w1,2(s1,IR2n) ___, L2(S1,IR2n)
This is an unbounded self-adjoint operator on the Hilbert space H = L^2 ( S^1 , IR^2 n)
with domain W = W^1 ·^2 (S^1 ,IR^2 n). The assumption det(Il - '11(1)) # 0 guarantees
that A is invertible, i.e. 0 is not an eigenvalue. Hence there is a splitting
H = E+ EBE-into the positive and negative eigenspaces of A. Denote A± = AIE± and denote by
p ± : H ___, E± the orthogonal projections. The operator -A+ generates a strongly
continuous semigroup of operators on E+ and A - generates a strongly continuous
semigroup of operators on E-. Denote these semigroups bys r-; e-A+s ands r-;
eA-s, respectively, where both are defined for s?: 0. Now define K: IR___, .C(H) by
{
e -A+s p + for s?: 0,
K(s) = '-e-A-sp-, for s < 0.
This function is discontinuous at s = 0, strongly continuous for s # 0, and satisfies
(18) llK(s)ll.ccHJ :::; e-6lslfor some constant o > 0. Consider the operator Q : L^2 (IR, H) ___, W^1 ·^2 (IR, H) n
L^2 (IR, W) defined by
Q'T/(s) = 1-: K(s -T)'f/(T) dT
for 'f/ E L^2 (R , H). We claim that this is the inverse of D. To see this note that