Fredholm Pairs Research Articles

Moduli Spaces of Singular Yamabe Metrics

Complete, conformally flat metrics of constant positive scalar curvature on the complement of k k points in the n n -sphere, k ≥ 2 k \ge 2 , n ≥ 3 n \ge 3 , were constructed by R. Schoen in 1988. We consider the problem of determining the moduli space of all such metrics. All such metrics are asymptotically periodic, and we develop the linear analysis necessary to understand the nonlinear problem. This includes a Fredholm theory and asymptotic regularity theory for the Laplacian on asymptotically periodic manifolds, which is of independent interest. The main result is that the moduli space is a locally real analytic variety of dimension k k . For a generic set of nearby conformal classes the moduli space is shown to be a k k -dimensional real analytic manifold. The structure as a real analytic variety is obtained by writing the space as an intersection of a Fredholm pair of infinite dimensional real analytic manifolds.

On pairs of projections in a Hilbert space

We give an algebraic derivation of the canonical form of a generic pair of projections. The result is used to determine the spectral shift of a pair of projections and various properties of Fredholm pairs.

Fredholm and invertible 𝑛-tuples of operators. The deformation problem

Using J. L. Taylor’s definition of joint spectrum, we study Fredholm and invertible n n -tuples of operators on a Hilbert space. We give the foundations for a "several variables" theory, including a natural generalization of Atkinson’s theorem and an index which well behaves. We obtain a characterization of joint invertibility in terms of a single operator and study the main examples at length. We then consider the deformation problem and solve it for the class of almost doubly commuting Fredholm pairs with a semi-Fredholm coordinate.