Related papers: The Index Bundle for Selfadjoint Fredholm Operator…

Naively, the analytic index of a family of self-adjoint Fredholm operators ought to be (an equivalence class of) the family of the kernels of these operators. The present paper is devoted to a rigorous version of this idea based on ideas of…

We associate to a parametrized family $f$ of nonlinear Fredholm maps possessing a trivial branch of zeroes an {\it index of bifurcation} $\beta(f)$ which provides an algebraic measure for the number of bifurcation points from the trivial…

This paper is devoted to Fredholm realizations of semi-Fredholm operators in a Hilbert space. Such a realization is determined by an abstract boundary condition, which is a subspace of the space of abstract boundary values. We find the…

In this paper, we define an analytical index for a continuous family of Fredholm operators parameterized by a topological space $\mathbb{X}$ into a Hilbert space $H,$ as a sequence of integers, extending naturally the usual definition of…

We show that every graph continuous family of unbounded operators in a Hilbert space becomes Riesz continuous after one-sided multiplication by an appropriate family of unitary operators. This result provides a simple definition of the…

Let $\Y$ be a smooth connected manifold, $\Sigma\subset\C$ an open set and $(\sigma,y)\to\scrP_y(\sigma)$ a family of unbounded Fredholm operators $D\subset H_1\to H_2$ of index 0 depending smoothly on $(y,\sigma)\in \Y\times \Sigma$ and…

This paper concerns Fredholm theory in several variables, and its applications to Hilbert spaces of analytic functions. One feature is the introduction of ideas from commutative algebra to operator theory. Specifically, we introduce a…

Consider a selfadjoint unbounded operator D on a Hilbert space H and a one parameter norm continuous family of selfadjoint bounded operators {A(t)} parametrized by the real line. Then under certain conditions \cite{RS95} that include the…

Heteroclinic orbits for one-parameter families of nonautonomous vectorfields appear in a very natural way in many physical applications. Inspired by some recent bifurcation results for homoclinic trajectories of nonautonomous vectorfield,…

We consider homoclinic solutions for Hamiltonian systems in symplectic Hilbert spaces and generalise spectral flow formulas that were proved by Pejsachowicz and the author in finite dimensions some years ago. Roughly speaking, our main…

We study the problem when an almost commuting $n$-tuple self-adjoint operators in an infinite dimensional separable Hilbert space $H$ is close to an $n$-tuple of commuting self-adjoint operators on $H.$ We give an affirmative answer to the…

We develop a $K$-theoretic approach to multiparameter bifurcation theory of homoclinic solutions of discrete non-autonomous dynamical systems from a branch of stationary solutions. As a byproduct we obtain a family index theorem for…

We show how the Atiyah-Singer family index theorem for both, usual and self-adjoint elliptic operators fits naturally into the framework of the Madsen-Tillmann-Weiss spectra. Our main theorem concerns bundles of odd-dimensional manifolds.…

A bounded operator $T$ on a separable, complex Hilbert space is said to be odd symmetric if $I^*T^tI=T$ where $I$ is a real unitary satisfying $I^2=-1$ and $T^t$ denotes the transpose of $T$. It is proved that such an operator can always be…

Almost flat finitely generated projective Hilbert C*-module bundles were successfully used by Hanke and Schick to prove special cases of the Strong Novikov Conjecture. Dadarlat later showed that it is possible to calculate the index of a…

We study the index theory of hypoelliptic operators on Carnot manifolds -- manifolds whose Lie algebra of vector fields is equipped with a filtration induced from sub-bundles of the tangent bundle. A Heisenberg pseudodifferential operator,…

If a differential operator $D$ on a smooth Hermitian vector bundle $S$ over a compact manifold $M$ is symmetric, it is essentially self-adjoint and so admits the use of functional calculus. If $D$ is also elliptic, then the Hilbert space of…

We define the equivariant family index of a family of elliptic operators invariant with respect to the free action of a bundle $\GR$ of Lie groups. If the fibers of $\GR \to B$ are simply-connected solvable, we then compute the Chern…

We study the parametrized Hamiltonian action functional for finite-dimensional families of Hamiltonians. We show that the linearized operator for the $L^2$-gradient lines is Fredholm and surjective, for a generic choice of Hamiltonian and…

Withdrawn due to a likely error with the homeomorphism at line (4). Old abstract: In the monograph 'Limit Operators and their Applications in Operator Theory', the authors define the operator spectrum of a band-dominated operator T and…