On the space of Fredholm operators
نویسنده
چکیده
We compare various topologies on the space of (possibly unbounded) Fredholm selfadjoint operators and explain their K-theoretic relevance.∗ Introduction The work of Atiyah and Singer on the index of elliptic operators on manifolds has singled out the role of the space of bounded Fredholm operators in topology. It is a classifying space for a very useful functor, the topological K-theory. This means that a continuous family (Lx)x∈X of elliptic pseudo differential operators parameterized by a compact CW -complex X naturally defines an element in the group K(X), the index of the family. In most examples, the elliptic operators are not bounded operators and thus the notion of continuity has to be defined carefully. The operator theorists have come up with a quick fix. The family x 7→ Lx of Fredholm operators is called Riesz continuous if and only if the families of bounded operators x 7→ Lx(1 + L ∗ xLx) , x 7→ L∗x(1 + LxL ∗ x) −1/2 are continuous with respect to the operator norm. In concrete applications this approach can be a nuisance. For example, consider as in [6] a Floer family of elliptic boundary value problems (parameterized by s ∈ S1) u(t) : [0, 1] → C, s ∈ [0, 2π]
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