You probably already know this, but the standard method is to assume there is a spectral gap, i.e. a real number not in the spectrum of $D_beta$ for all $beta$. So although, as you point out, zero might not work, another real number might, or more generally, if there is a function $g:Bto R$ so that $g(beta)$ does not lie in the spectrum of $D_beta$ then you can use it to construct a spectral section.
Such functions can always be found locally, and so you can try to patch these local solutions; perhaps this is what underlies Melrose-Piazza.
Another general method, when the space $B$ is an interval and the dependence of $D_beta$ on $beta$ is analytic, is to use the Kato selection lemma, which allows you to analytically continue eigenvectors, and in particular if you take the positive eigenspan at one parameter, you can extend this to a spectral section over the interval, even if there is no spectral gap. Presumably if you are careful you can extend this idea for more general analytic families. Typically families which arise in geometric contexts vary analytically, although this is not always easy to check.
No comments:
Post a Comment