Fredholm differential operators with unbounded coefficients

We prove that a first-order linear differential operator G with unbounded operator coefficients is Fredholm on spaces of functions on with values in a reflexive Banach space if and only if the corresponding strongly continuous evolution family has exponential dichotomies on both and and a pair of the ranges of the dichotomy projections is Fredholm, and that the Fredholm index of G is equal to the Fredholm index of the pair. The operator G is the generator of the evolution semigroup associated with the evolution family. In the case when the evolution family is the propagator of a well-posed differential equation u′(t)=A(t)u(t) with, generally, unbounded operators , the operator G is a closure of the operator . Thus, this paper provides a complete infinite-dimensional generalization of well-known finite-dimensional results by Palmer, and by Ben-Artzi and Gohberg.     

Fredholm property of non-smooth pseudodifferential operators

In this paper we prove sufficient conditions for the Fredholm property of a non-smooth pseudodifferential operator P which symbol is in a Hölder space with respect to the spatial variable. As a main ingredient for the proof we use a suitable symbol-smoothing.     

Nonhomogeneous dirichlet problem for elliptic operators with axially discontinuous coefficients

Si studia, in un cilindro, il problema di Dirichlet per l'equazione ellittica del II ordine: $\text{L}{}_{\mathrm{\alpha }}\text{u = ?}$, dove $\text{L}{}_{\mathrm{\alpha }}\text{=}\text{αΔ + (1 ? 3α)∑}{}_{\mathrm{ij\; =\; 1}}{}^2\text{x}{}_{\mathrm{i}}\text{x}{}_{\mathrm{j}}\text{(x}{}_1{}^2\text{+ x}{}_2{}^2\text{)}{}^{\mathrm{?1}}\text{?}{}^2\text{?x}{}_{\mathrm{i}}\text{?x}{}_{\mathrm{j}}\text{, α ? (0,}\text{1}\text{3}\text{]}$è l'operatore a coefficienti discontinui sull'asse x₃ già introdotto da N. Ural'tseva per mostrare che l'equazione considerata può non avere soluzione nello spazio di Sobolev W^2,p(p > 2) per qualche f?L^p. In questo lavoro si danno limitazioni a priori e teoremi di esistenza e unicità in W^2,p quando p varia in un intervallo (p₁(α), p₂(α)), dipendente dalla costante di ellitticità α. Se p = p₂(α) le limitazioni a priori cadono: l'esempio è quello di Ural'tseva.     

Band-dominated operators with operator-valued coefficients, their Fredholm properties and finite sections

The central theme of the present paper are band and band-dominated operators, i.e. norm limits of band operators. In the first part, we generalize the results from [24] and [25] concerning the Fredholm properties of band-dominated operators and the applicability of the finite section method to the case of operators with operator-valued coefficients. We characterize these properties in terms of the limit operators of the given band-dominated operator. The main objective of the second part is to apply these results to pseudodifferential operators on cones in ⁿ which is possible after a suitable discretization.Partially supported by the German Research Foundation (DFG) under Grant Nr. 436 RUS 17/67/98.     

Uniqueness in the dirichlet problem for some elliptic operators with discontinuous coefficients

Uniqueness is proved for the Dirichlet problem for second order nondivergence form elliptic operators with coefficients continuous except at a countable set of points having at most one accumulation point. Moreover, gradient estimates are proved.The authors are partially supported by the National Science Foundation Grant no. NSF/DMS 8421377-04.     

Fredholm composition operators

Fredholm composition operators on a variety of Hilbert spaces of analytic functions on domains in , are characterized.

     

Fredholm property of a family of operators on the five-dimensional Heisenberg group

An expansion in Euclidean spherical harmonics on the ball in the Heisenberg group of dimension five decomposes the Dirichlet problem for the Laplacian in an infinite number of two-dimensional problems. Fundamental solutions are obtained for each of the partial differential operators in these problems, thus reducing them further (via layer potentials) to one-dimensional integral equations. The main result in this article states that the corresponding integral operators are Fredholm in appropriate weighted L² spaces.     

Fredholm weighted composition operators

We characterize the Fredholm weighted composition operators onC(X). In particular, ifX is a set with some regular property like intervals or balls inR ⁿ , our characterization implies that a weighted composition operator is Fredholm if and only if it is invertible. This equivalence is true for weighted composition operators onL ^p (), where is a nonatomic measure (1p<).     

Forward-backward SDEs with discontinuous coefficients

In this paper, we are interested in the well-posedness of a class of fully coupled forward-backward SDE (FBSDE) in which the forward drift coefficient is allowed to be discontinuous with respect to the backward component of the solution. Such an FBSDE is motivated by a practical issue in regime-switching term structure interest rate models, and the discontinuity makes it beyond any existing framework of FBSDEs. In a Markovian setting with non-degenerate forward diffusion, we show that a decoupling function can still be constructed and that it is a Sobolev solution to the corresponding quasilinear PDE. As a consequence we can then argue that the FBSDE admits a weak solution in the sense of [1 Antonelli, F., Ma, J. (2003). Weak solutions of forward-backward SDE’s. Stochastic Analysis and Applications 21(3):493–514.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 2 Ma, J., Zhang, J., Zheng, Z. (2008). Weak solutions for backward stochastic differential equations, A martingale approach. The Annals of Probability 36(6):2092–2125.[Crossref], [Web of Science ®] , [Google Scholar]]. In the one-dimensional case, we further prove that the weak solution of the FBSDE is actually strong, and it is pathwisely unique. Our approach does not use the well-known Yamada–Watanabe Theorem, but instead follows the idea of Krylov for SDEs with measurable coefficients.     

Perturbing Fredholm operators to obtain invertible operators

A condition is given on a set $\text{Ol}$ of operators on Hilbert space that guarantees it has the following property: For any Fredholm operator T of index zero there exists anA?$\text{A}$ such that T + ?A is invertible for all sufficiently small nonzero ?. As a corollary one obtains in a quite general setting the density of the invertible Toeplitz operators in the set of Fredholm Toeplitz operators of index zero.