A note on generalized invertibility and invariants of infinite matrices

The classes of band-dominated operators and the subclass of operators in the Wiener algebra $${\mathcal {W}}$$ W are known to be inverse closed. This paper studies and extends partially known results of that type for one-sided and generalized invertibility. Furthermore, for the operators in the Wiener algebra $${\mathcal {W}}$$ W invertibility, the Fredholm property and the Fredholm index are known to be independent of the underlying space $$l^p$$ l p , $$1\le p\le \infty $$ 1 ≤ p ≤ ∞ . Here this is completed by the observation that even the kernel and a suitable direct complement of the range as well as generalized inverses of operators in $${\mathcal {W}}$$ W are invariant w.r.t. p.

Analysis of gauged Witten equation

The gauged Witten equation was essentially introduced by Witten in his formulation of the gauged linear σ-model (GLSM), which explains the so-called Landau–Ginzburg/Calabi–Yau correspondence. This is the first paper in a series towards a mathematical construction of GLSM, in which we set up a proper framework for studying the gauged Witten equation and its perturbations. We also prove several analytical properties of solutions and moduli spaces of the perturbed gauged Witten equation. We prove that solutions have nice asymptotic behavior on cylindrical ends of the domain. Under a good perturbation scheme, the energies of solutions are shown to be uniformly bounded by a constant depending only on the topological type. We prove that the linearization of the perturbed gauged Witten equation is Fredholm, and we calculate its Fredholm index. Finally, we define a notion of stable solutions and prove a compactness theorem for the moduli space of solutions over a fixed domain curve.

Essential Spectrum and Fredholm Index for Certain Composition Operators

We investigate a composition operator on $H^\infty(U)$, $U$ a subdomain of the open unit disk, for which the essential resolvent has infinitely many components, and for which the Fredholm index of the resolvent operator attains all nonnegative integer values.

A topological index theorem for manifolds with corners

We define an analytic index and prove a topological index theorem for a non-compact manifold M0 with poly-cylindrical ends. Our topological index theorem depends only on the principal symbol, and establishes the equality of the topological and analytical index in the group K0(C*(M)), where C*(M) is a canonical C*-algebra associated to the canonical compactification M of M0. Our topological index is thus, in general, not an integer, unlike the usual Fredholm index appearing in the Atiyah–Singer theorem, which is an integer. This will lead, as an application in a subsequent paper, to the determination of the K-theory groups K0(C*(M)) of the groupoid C*-algebra of the manifolds with corners M. We also prove that an elliptic operator P on M0 has an invertible perturbation P+R by a lower-order operator if and only if its analytic index vanishes.