Finite Dimensional Kernel Research Articles

Linearized Boltzmann Collision Operator: I. Polyatomic Molecules Modeled by a Discrete Internal Energy Variable and Multicomponent Mixtures

The linearized Boltzmann collision operator appears in many important applications of the Boltzmann equation. Therefore, knowing its main properties is of great interest. This work extends some classical results for the linearized Boltzmann collision operator for monatomic single species to the case of polyatomic single species, while also reviewing corresponding results for multicomponent mixtures of monatomic species. The polyatomicity is modeled by a discrete internal energy variable, that can take a finite number of (given) different values. Results concerning the linearized Boltzmann collision operator being a nonnegative symmetric operator with a finite-dimensional kernel are reviewed.A compactness result, saying that the linearized operator can be decomposed into a sum of a positive multiplication operator, the collision frequency, and a compact operator, bringing e.g., self-adjointness, is extended from the classical result for monatomic single species, under reasonable assumptions on the collision kernel. With a probabilistic formulation of the collision operator as a starting point, the compactness property is shown by a splitting, such that the terms can be shown to be, or be the uniform limit of, Hilbert-Schmidt integral operators and as such being compact operators. Moreover, bounds on - including coercivity of - the collision frequency are obtained for a hard sphere like model, from which Fredholmness of the linearized collision operator follows, as well as its domain.

Resolvents for fractional-order operators with nonhomogeneous local boundary conditions

For 2a-order strongly elliptic operators P generalizing (−Δ)a, 0<a<1, the homogeneous Dirichlet problem on a bounded open set Ω⊂Rn has been widely studied. Pseudodifferential methods have been applied by the present author when Ω is smooth; this is extended in a recent joint work with Helmut Abels showing exact regularity theorems in the scale of Lq-Sobolev spaces Hqs for 1<q<∞, when Ω is Cτ+1 with a finite τ>2a. We now develop this into existence-and-uniqueness theorems (or Fredholm theorems), by a study of the Lp-Dirichlet realizations of P and P⁎, showing that there are finite-dimensional kernels and cokernels lying in daCα(Ω‾) with suitable α>0, d(x)=dist(x,∂Ω). Similar results are established for P−λI, λ∈C. The solution spaces equal a-transmission spaces Hqa(t)(Ω‾).Moreover, the results are extended to nonhomogeneous Dirichlet problems prescribing the local Dirichlet trace (u/da−1)|∂Ω. They are solvable in the larger spaces Hq(a−1)(t)(Ω‾). Furthermore, the nonhomogeneous problem with a spectral parameter λ∈C,Pu−λu=f in Ω,u=0 in Rn∖Ω,(u/da−1)|∂Ω=φ on ∂Ω, is for q<(1−a)−1 shown to be uniquely resp. Fredholm solvable when λ is in the resolvent set resp. the spectrum of the L2-Dirichlet realization.The results open up for applications of functional analysis methods. Here we establish solvability results for evolution problems with a time-parameter t, both in the case of the homogeneous Dirichlet condition, and the case where a nonhomogeneous Dirichlet trace (u(x,t)/da−1(x))|x∈∂Ω is prescribed.

Cauchy Spinors on 3-Manifolds

Let $$\mathcal {Z}$$ be a spin 4-manifold carrying a parallel spinor and $$M\hookrightarrow \mathcal {Z}$$ a hypersurface. The second fundamental form of the embedding induces a flat metric connection on TM. Such flat connections satisfy a non-elliptic, non-linear equation in terms of a symmetric 2-tensor on M. When M is compact and has positive scalar curvature, the linearized equation has finite-dimensional kernel. Four families of solutions are known on the round 3-sphere $${\mathbb {S}}^3$$ . We study the linearized equation in the vicinity of these solutions and we construct as a byproduct an incomplete hyperkähler metric on $${\mathbb {S}}^3\times \mathbb {R}$$ closely related to the Euclidean Taub-NUT metric on $$\mathbb {R}^4$$ . On $${\mathbb {S}}^3$$ there do not exist other solutions which either are constant in a left (or right) invariant frame, have three distinct constant eigenvalues, or are invariant in the direction of a left (or right)-invariant eigenvector. We deduce from this last result an extension of Liebmann’s sphere rigidity theorem.

First-Kind Boundary Integral Equations for the Dirac Operator in 3-Dimensional Lipschitz Domains

We develop novel first-kind boundary integral equations for Euclidean Dirac operators in 3D Lipschitz domains comprising square-integrable potentials and involving only weakly singular kernels. Generalized Garding inequalities are derived and we establish that the obtained boundary integral operators are Fredholm of index zero. Their finite dimensional kernels are characterized and we show that their dimension is equal to the number of topological invariants of the domain's boundary, in other words to the sum of its Betti numbers. This is explained by the fundamental discovery that the associated bilinear forms agree with those induced by the 2D surface Dirac operators for H-1/2 surface de Rham Hilbert complexes whose underlying inner-products are the non-local inner products defined through the classical single-layer boundary integral operators for the Laplacian. Decay conditions for well-posedness in natural energy spaces of the Dirac system in unbounded exterior domains are also presented.

Unbounded Toeplitz Operators with Rational Symbols

Unbounded (and bounded) Toeplitz operators (TO) with rational symbols are analysed in detail showing that they are densely defined closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains, ranges, spectral and Fredholm points are determined. In particular, in the symmetric case, i.e., for a real rational symbol the deficiency spaces and indices are explicitly available. The concluding section gives a brief overview on the research on unbounded TO in order to locate the present contribution. Regarding properties of unbounded TO in general, it furnishes some new results recalling the close relationship to Wiener-Hopf operators and, in case of semiboundedness, to singular operators of Hilbert transformation type. Specific symbols considered in the literature admit further analysis. Some conclusions are drawn for semibounded integrable and real square-integrable symbols. There is an approach to semibounded TO, which starts from closable semibounded forms related to a Toeplitz matrix. The Friedrichs extension of the TO associated with such a form is studied. Finally, analytic TO and Toeplitz-like operators are briefly examined, which in general differ from the TO treated here.

Unbounded Wiener\u2013Hopf Operators and Isomorphic Singular Integral Operators

Some basics of a theory of unbounded Wiener–Hopf operators (WH) are developed. The alternative is shown that the domain of a WH is either zero or dense. The symbols for non-trivial WH are determined explicitly by an integrability property. WH are characterized by shift invariance. We study in detail WH with rational symbols showing that they are densely defined, closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains, ranges, spectral and Fredholm points are explicitly determined. Another topic concerns semibounded WH. There is a canonical representation of a semibounded WH using a product of a closable operator and its adjoint. The Friedrichs extension is obtained replacing the operator by its closure. The polar decomposition gives rise to a Hilbert space isomorphism relating a semibounded WH to a singular integral operator of Hilbert transformation type. This remarkable relationship, which allows to transfer results and methods reciprocally, is new also in the thoroughly studied case of bounded WH.

Clark Representation for Local Times of Self-Intersection of Gaussian Integrators

We prove the existence of multiple local times of self-intersection for a class of Gaussian integrators generated by operators with finite-dimensional kernels,, describe its Ito–Wiener expansion and establish the Clark representation.

Subspaces of Banach spaces with big slices

We study when diameter two properties pass down to subspaces. We obtain that the slice two property (respectively diameter two property, strong diameter two property) passes down from a Banach space $X$ to a subspace $Y$ whenever $Y$ is complemented by a norm one projection with finite-dimensional kernel (respectively the quotient $X/Y$ is finite dimensional, $X/Y$ is strongly regular). Also we study the same problem for dual properties of the above ones, as having octahedral, weakly octahedral or 2-rough norm.

Stability analysis in magnetic resonance elastography II

We consider the inverse problem of finding unknown elastic parameters from internal measurements of displacement fields for tissues. In the sequel to [5], we use pseudodifferential methods for the problem of recovering the shear modulus for Stokes systems from internal data. We prove stability estimates in d=2,3 with reduced regularity on the estimates and show that the presence of a finite dimensional kernel can be removed. This implies the convergence of the Landweber numerical iteration scheme. We also show that these hypotheses are natural for experimental use in constructing shear modulus distributions.

Finite-dimensional representations of the elliptic modular double

We investigate the kernel space of an integral operator M(g) depending on the “spin” g and describing an elliptic Fourier transformation. The operator M(g) is an intertwiner for the elliptic modular double formed from a pair of Sklyanin algebras with the parameters η and τ, Imτ > 0, Imη > 0. For two-dimensional lattices g = nη + mτ/2 and g = 1/2 + nη + mτ/2 with incommensurate 1, 2η, τand integers n,m > 0, the operator M(g) has a finite-dimensional kernel that consists of the products of theta functions with two different modular parameters and is invariant under the action of generators of the elliptic modular double.

On intersections of ranges of projections of norm one in Banach spaces

In this short note we are interested in studying Banach spaces in which the range of a projection of norm one whose kernel is of finite dimension is the intersection of ranges of finitely many projections of norm one whose kernels are of dimension one. We show that for a certain class of Banach spaces X X , the natural duality between X X and X ∗ ∗ X^{\ast \ast } can be exploited when the range of the projection is of finite codimension. We show that if X ∗ X^\ast is isometric to L 1 ( μ ) L^1(\mu ) , then any central subspace of finite codimension is an intersection of central subspaces of codimension one. These results extend a recent result of Bandyopadhyay and Dutta which was proved for ranges of projections of norm one with finite dimensional kernel in continuous function spaces and unifies some earlier work of Baronti and Papini.

Variational Methods for NLEV Approximation Near a Bifurcation Point

We review some more and less recent results concerning bounds on nonlinear eigenvalues (NLEV) for gradient operators. In particular, we discuss the asymptotic behaviour of NLEV (as the norm of the eigenvector tends to zero) in bifurcation problems from the line of trivial solutions, considering perturbations of linear self-adjoint operators in a Hilbert space. The proofs are based on the Lusternik-Schnirelmann theory of critical points on one side and on the Lyapounov-Schmidt reduction to the relevant finite-dimensional kernel on the other side. The results are applied to some semilinear elliptic operators in bounded domains ofℝN. A section reviewing some general facts about eigenvalues of linear and nonlinear operators is included.

Coercive estimates and integral representation formulas on carnot groups

We establish Sobolev-type integral representation formulas for differential functions on Carnot groups. We prove coercive estimates for homogeneous differential operators with constant coefficients and finite-dimensional kernel. These integral representation formulas are new in the Euclidean spaces as well.

Geometric structures on loop and path spaces

The loop space associated to a Riemannian manifold admits a quasisymplectic structure (that is, a closed 2-form which is non-degenerate up to a finite-dimensional kernel). We show how to construct a compatible almost-complex structure. Finally conditions to have contact structures on loop spaces are studied.

A Characterization of Schauder Frames Which Are Near-Schauder Bases

A basic problem of interest in connection with the study of Schauder frames in Banach spaces is that of characterizing those Schauder frames which can essentially be regarded as Schauder bases. In this paper, we give a solution to this problem using the notion of the minimal-associated sequence spaces and the minimal-associated reconstruction operators for Schauder frames. We prove that a Schauder frame is a near-Schauder basis if and only if the kernel of the minimal-associated reconstruction operator contains no copy of $c_0$. In particular, a Schauder frame of a Banach space with no copy of $c_0$ is a near-Schauder basis if and only if the minimal-associated sequence space contains no copy of $c_0$. In these cases, the minimal-associated reconstruction operator has a finite dimensional kernel and the dimension of the kernel is exactly the excess of the near-Schauder basis. Using these results, we make related applications on Besselian frames and near-Riesz bases.

Solvability conditions for elliptic problems with non-Fredholm operators

The paper is devoted to solvability conditions for linear elliptic problems with non-Fredholm operators. We show that the operator becomes normally solvable with a finite-dimensional kernel on properly chosen subspaces. In the particular case of a scalar

Maximal, Minimal, and Primary Invariant Subspaces

Let X be a complex infinite dimensional Banach space. An operator L on X is called of subcritical class, if ∑∞n=1n−3/2log+‖Ln‖<∞. Assume that T is an operator on X whose iterates have norms of polynomial growth. We prove that if T has a range of finite codimension and a left inverse of subcritical class, then every maximal invariant subspace of T has codimension one, and if T has a finite dimensional kernel and a right inverse of subcritical class, then every minimal invariant subspace of T is one dimensional. Using these results we obtain new information on the invariant subspace lattices of shifts and backward shifts on a wide class of Banach spaces of analytic functions on the unit disc. We also introduce the notion of primary invariant subspaces, and determine their structure for a large class of shifts.

Dynamical entropy for systems with stochastic perturbation

Dynamics of deterministic systems perturbed by random additive noise is characterized quantitatively. Since for such systems the Kolmogorov-Sinai (KS) entropy diverges if the diameter of the partition tends to zero, we analyze the difference between the total entropy of a noisy system and the entropy of the noise itself. We show that this quantity is finite and non-negative and we call it the dynamical entropy of the noisy system. In the weak noise limit this quantity is conjectured to tend to the KS entropy of the deterministic system. In particular, we consider one-dimensional systems with noise described by a finite-dimensional kernel for which the Frobenius-Perron operator can be represented by a finite matrix.

Solvability of semilinear abstract equations at resonance

We establish a generalization of the Cesari-Kannan existence result for problems of the type Lx = N(x), x∈X where X is a separable Hilbert functional space, L is a selfadjoint linear differential operator with nontrivial finite dimensional kernel and N:X→X is a bounded continuous nonlinear operator. This generalization leads to new results when the dimension of the kernel of L is greater than one. Applications to systems of second order ordinary differential equations are given.

Functional Persistence of Excitation and Observability for Learning Control Systems

Adaptive systems involving function learning can be formulated in terms of integral equations of the first kind, possibly with separable, finite-dimensional kernels. The learning process involves estimating the influence functions (Messner et al., 1989). To achieve convergence of the influence function estimates and exponentially stability, it is important to have persistence of excitation in the training tasks. This paper develops the concept of functional persistence of excitation (PE), and the associated concept of functional uniform complete observability (UCO). Relevant PE and UCO properties for linear systems are developed. For example, a key result is that uniform complete observability in this context is maintained under bounded integral operator output injection—a natural generalization of the corresponding finite dimensional result. This paper also demonstrates the application of the theory to linear error equations associated with a repetitive control algorithm.