A splitting theorem

Introduction 1 Some analysis in Hilbert spaces and scales 1.1 Smooth and analytic maps 1.2 Scales of Hilbert spaces and interpolation 1.3 Differential forms 2 Symplectic structures and Hamiltonian equations 2.1 Basic definitions 2.2 Symplectic transformations 2.3 Darboux lemmas Appendix. Time-quasiperiodic solutions 3 Lax-integrable Hamiltonian equations and their integrable subsystems 3.1 Examples of Hamiltonian PDEs 3.2 Lax-integrable equations 3.3 Integrable subsystems 4 Finite-gap manifolds and theta-formulas 4.1 Finite-gap manifolds for the KdV equation 4.2 The Its-Matveev theta-formulas 4.3 Higher equations from the KdV hierarchy 4.4 Sine-Gordon equation under Dirichlet boundary conditions 5 Linearised equations and their Floquet solutions 5.1 The linearised equation 5.2 Floquet solutions 5.3 Complete systems of Floquet solutions 5.4 Lower-dimensional invariant tori of finite-dimensional systems and Floquet’s theorem 6 Linearised Lax-integrable equations 6.1 Abstract situation 6.2 Linearised KdV equation 6.3 Higher KdV-equations 6.4 Linearised SG equation 7 Normal form 7.1 A normal form theorem 7.2 Examples 8 The KAM theorem 8.1 The main theorem and related results 8.2 Reduction to a parameter-depending case 8.3 A KAM-theorem for parameter-depending equations 8.4 Completion of the Main Theorem’s proof (Step 4) 9 Examples 9.1 Perturbed KdV equation 9.2 Higher KdV equations 9.3 Perturbed SG equation 9.4 KAM-persistence of lower-dimensional invariant tori of nonlinear finite-dimensional systems References

Trace ideal criteria for Toeplitz operators

Symplectic structures and hamiltonian systems in scales of hilbert spaces

We consider semilinear evolution equations for which the linear part is normal and generates a strongly continuous semigroup, and the nonlinear part is sufficiently smooth on a scale of Hilbert spaces. We approximate their semi-flow by an implicit A-stable Runge–Kutta discretization in time and a spectral Galerkin truncation in space. We show regularity of the Galerkin-truncated semi-flow and its time discretization on open sets of initial values with bounds that are uniform in the spatial resolution and the initial value. We also prove convergence of the space-time discretization without any condition that couples the time step to the spatial resolution. We then estimate the Galerkin truncation error for the semi-flow of the evolution equation, its Runge–Kutta discretization and their respective derivatives, showing how the order of the Galerkin truncation error depends on the smoothness of the initial data. Our results apply, in particular, to the semilinear wave equation and to the nonlinear Schrodinger equation.

G. Point interactions and scales of Hilbert spaces

The celebrated Cheeger-Gromoll splitting theorem (see [ChGr], and also [EsHe, Wu]) implies that a connected, complete Riemannian manifold having nonnegative Ricci curvature and two ends is isometric to a Riemannian product, where one factor is a line. There is a version of this theorem for compact, connected Riemannian manifolds M with two smooth boundary components. It states that if M has nonnegative Ricci curvature and two boundary components, both of which have nonnegative mean curvature with respect to their inward normals, then M is isometric to the Riemannian product of N1 with a closed interval (in particular N1 is isometric to N2). In this paper we will prove an analog of this result for manifolds with Ricci curvature ≥ −(n − 1). Since our argument parallels the proof of the splitting theorem for manifolds with nonnegative Ricci curvature, we present the proofs simultaneously by letting δ be either zero or one in what follows.

Scale space representations over Euclidean space are closely related to continuous wavelet transforms. E.g., taking the time derivative of a Gaussian scale space representation yields a continuous wavelet transform with respect to the Mexican Hat wavelet. Departing from this observation, we introduce a continuous diffusion wavelet transform associated to any diffusion semigroup acting on an abstract Hilbert space. Under quite general assumptions this wavelet transform is norm-preserving, thus giving rise to wavelet-type inversion formulae. For the diffusion associated to the heat equation on the Heisenberg group ℍ, we recover the so-called Mexican Hat wavelet previously obtained by Mayeli. Our approach provides an alternative short proof of admissibility for the Mexican Hat wavelet on ℍ, as well as a new family of wavelets for ℍ and more general homogeneous Lie groups.

The known proofs of Keller's theorem that any infinite-dimensional compact convex set in Hilbert space is homeomorphic to the Hilbert cube are analytic. Here a topological proof of this theorem is given. A new approach to the old theorem leads to a proof of an equivariant version of Keller's theorem.

When the singular finite rank perturbations of an unbounded selfadjoint operator Ao in a Hilbert space c, formally defined by A(α) = A 0 + GαG*, are lifted to an exit Pontryagin space ℌ by means of an operator model, they become ordinary range perturbations of a self-adjoint operator H ∞ in ℌ ⊃ ℌ0:H T = H ∞-ΩT -1Ω*. Here G is a mapping from ℂd into some scale space ℌ (A 0), ℕ, of generalized elements associated with A 0, while Ω is a mapping from into ℂd the extended space ℌ, where H T , is defined. The connection between these two perturbation formulas is studied.KeywordsSingular finite rank perturbationextension theoryKrein’s formulaboundary tripletWeyl functiongeneralized Nevanlinna functionoperator model

Inner products of Sobolev type are extremely useful for image reconstruction of images from a sparse set of α-scale space features. The common (non)-linear reconstruction frameworks, follow an Euler Lagrange minimization. If the Lagrangian (prior) is a norm induced by an inner product of a Hilbert space, this Euler Lagrange minimization boils down to a simple orthogonal projection within the corresponding Hilbert space. This basic observation has been overlooked in image analysis for the cases where the Lagrangian equals a norm of Sobolev type, resulting in iterative (non-linear) numerical methods, where already an exact solution with non-iterative linear algorithm is at hand. Therefore we provide a general theory on linear image reconstructions and metameric classes of images. By applying this theory we obtain visually more attractive reconstructions than the previously proposed linear methods and we find connected curves in the metameric class of images, determined by a fixed set of linear features, with a monotonic increase of smoothness. Although the theory can be applied to any linear feature reconstruction or principle component analysis, we mainly focus on reconstructions from so-called topological features (such as top-points and grey-value flux) in scale space, obtained from geometrical observations in the deep structure of a scale space.KeywordsScale SpaceSobolev SpacesGelfand TriplesTikhonov RegularizationTop Point ReconstructionDeep StructureFlux Features

In this chapter we shall introduce some of the fundamental theorems for manifolds with lower Ricci curvature bounds. Two important techniques will be developed: relative volume comparison and weak upper bounds for the Laplacian of distance functions. With these techniques we shall show numerous results on restrictions of fundamental groups of such spaces and also present a different proof of the estimate for the first Betti number by Bochner. The proof of the splitting theorem is self-contained. It uses the generalized maximum principle, but we show how one can get around the regularity issue for harmonic distance functions using some of our previous work on distance functions.

We give a new and relatively easy proof of the splitting theorem of the second author for the spaces in the Omega spectrum for BP. We then give the first published proofs of our similar theorems for the spectra P(n).

We introduce the space of equivariant local maps\nand present the full proof of the splitting theorem for the set of otopy\nclasses of such maps in the case of a representation of a compact Lie group.

We study a regular elliptic boundary-value problem for a homogeneous differential equation in a bounded domain. We prove that the operator of this problem is a Fredholm (Noether) operator in a two-sided improved scale of functional Hilbert spaces. The elements of this scale are Hormander-Volevich-Paneyakh isotropic spaces. We establish an a priori estimate for a solution and investigate its regularity.

A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time