We obtain an identity between Fredholm determinants of two kinds of operators, one acting on functions on the unit circle and the other acting on functions on a subset of the integers. This identity is a generalization of an identity between a Toeplitz determinant and a Fredholm determinant that has appeared in the random permutation context. Using this identity, we prove, in particular, conver...

We show that the existence of a Fredholm element of the zero calculus of pseudodifferential operators on a compact manifold with boundary with a given elliptic symbol is determined, up to stability, by the vanishing of the Atiyah-Bott obstruction. It follows that, up to small deformations and stability, the same symbols have Fredholm realizations in the zero calculus, in the scattering calculus...

*Correspondence: [email protected] 2Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia Full list of author information is available at the end of the article Abstract In this paper, we derive a new generalized Volterra–Fredholm integral inequality and use it to study the dependence of solutions on the ini...

We study the composition of time-frequency localization operators (wavepacket operators) and develop a symbolic calculus of such operators on modulation spaces. The use of time-frequency methods (phase space methods) allows the use of rough symbols of ultra-rapid growth in place of smooth symbols in the standard classes. As the main application it is shown that, in general, a localization opera...

Linear elliptic problems in bounded domains are normally solvable with a finite-dimensional kernel and a finite codimension of the image, that is, satisfy the Fredholm property, if the ellipticity condition, the condition of proper ellipticity and the Lopatinskii condition are satisfied. In the case of unbounded domains these conditions are not sufficient any more. The necessary and sufficient ...

The aim of this paper is twofold: On one hand we discuss an abstract approach to symmetrized Fredholm perturbation determinants and an associated trace formula for a pair of operators of positive-type, extending a classical trace formula. On the other hand, we continue a recent systematic study of boundary data maps, that is, 2× 2 matrix-valued Dirichlet-to-Neumann and more generally, Robin-to-...

1. Elliptic Differential Operators 1 1.1. Partial differential operators 1 1.2. Sobolev spaces and Hölder spaces 10 1.3. Apriori estimates and elliptic regularity 20 1.4. Elliptic operators on compact manifolds 28 2. Non-linear Elliptic Equations 35 2.1. Banach manifolds and Fredholm operators 35 2.2. Moduli space of nonlinear elliptic equations 39 2.3. The Sard-Smale theorem and transversality...

In this paper, Semi-orthogonal (SO) B-spline scaling functions and wavelets and their dual functions are presented to approximate the solutions of linear and non-linear second order Fredholm integro-differential equations. The B-spline scaling functions and wavelets, their properties and the operational matrices of derivative for this functions are presented to reduce the solution of linear and...

In the paper one considers the local structure of the Fredholm joint spectrum of commuting n-tuples of operators. A connection between the spatial characteristics of operators and the algebraic invariant of the corresponding coherent sheaves is investigated. A notion of Weyl joint spectrum of commuting n-tuple is introduced.