Local analytic splitting

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Remark . In [KK2] this theorem will be proven in the more difficult case when X has boundary (for operators of order 1) with Atiyah-Patodi-Singer boundary conditions which vary with t. See also [KKl].

We can apply this result to conclude that С and rj invariants of self-adjoint differential operators coupled to flat connections vary analytically with respect to the holonomy representation. We begin with a proposition:

5 . 2 Proposition. Let D,, for tG[0,1], be an analytic path of self-adjoint elliptic differential operators. Let {А„(0}Г=1 ^^ ^^^ corresponding analytic paths of values. Then the set

{ п|Я „ ( 0 = 0 for some tG[0,1]}

is finite.

Proof Suppose the proposition is false. Then there exists an increasing sequence «1, «2,. . . of positive integers and t^, ^2» • • • ^№' 1] ^o that:

1 . t^ converges monotonically to some t^elf), 1]

2 . Я„, = 0.

Since the kernel of D^ is finite dimensional for all t, we may assume, after takmg a subsequence, that either Я„/Го) > 0 for all i or that k^j^t) < 0 for all i. Assume that К (^o) > 0 for all i; the other case is similar.

' Let Я be a positive number smaller than the smallest non-zero eigenvalue of D,^. Since A„,(ti) = 0 and A„/to) > Я, it follows that there is an s^ between t^ and to so that X^(s,) = X. This contradicts the following standard lemma:

5 . 3 Lemma. Suppose that D^ = Dq + A, where D, is a path of elliptic self-adjoint operators of order d {with smooth coefficients) and A, is a path of operators of order strictly less than d. Ift„ is a sequence converging to to, and A„GSpec(D,^) converge to Я, then ÀeSpec{DfJ.

Proof We may assume that to = 0. Write D„ = D,^ and A„ = D„ + Do- Since Do is elliptic we can take the Lj norm to be \\t\\j = ||тр + l|l>o^f, where || • || denotes the L^ norm. Since A„ has lower order, it is bounded from Lj to L^, and so the operator norm of A„ from Lj to L^ norm converges to 0. We assume this norm is always less than i by passing to a subsequence.

Choose eigenvectors ф„ for D„ with L^ norm equal to 1. Then

Unfa = Unf + \\1>оФ\\'

= 1 + \\0„ф„-АМ\'

^i + A „ ^ + MJI'll0nL'.

Thus \\фJjйî{^+ ^n) and since À„ converges to A, the ф„ are L^bounded. Thus a subsequence converges to ф weakly in Lj and strongly in L^. Now since ф„ converges weakly in Lj to ф, we have

<Фп . T> + <ОоФп, Dot} -> <ф, t> + <Do^, Dot)

for any test function т (here <•, •> denotes the L^ inner product). Since ф„ verges to Ф in L^ it follows that <Do</>n. ^o^^) ^ <^оФп^ ^o^>' and since Dq is self-adjoint <Do0„, т> -^ <0оф„, т>, i.e. ОоФп converges weakly to ОоФ in L^.