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...

Let (U(t))t≥0 be a C0-semigroup of bounded linear operators on a Banach space X. In this paper, we establish that if, for some t0 > 0, U(t0) is a Fredholm (resp., semiFredholm) operator, then (U(t))t≥0 is a Fredholm (resp., semi-Fredholm) semigroup. Moreover, we give a necessary and sufficient condition guaranteeing that (U(t))t≥0 can be embedded in a C0-group on X. Also we study semigroups whi...

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.

Infinite-dimensional difference operators are studied. Under the assumption that the coefficients of the operator have limits at infinity, limiting operators and associated polynomials are introduced. Under some specific conditions on the polynomials, the operator is Fredholm and has the zero index. Solvability conditions are obtained and the exponential behavior of solutions of the homogeneous...

Based on the recently introduced uniform λ−adjustment for closed subspaces of Banach spaces we extend the concept of the strictly singular and nitely strictly singular operators to the sequences of closed subspaces and operators in Banach spaces and prove theorems about lower semi Fredholm stability. We also state some new open questions related to strict singularity and the geometry of Banach ...

A classical problem in operator theory has been to determine the spectrum of Toeplitz-like operators on Hilbert spaces of vector-valued holomorphic functions on the open unit ball in Cm. In this note we obtain necessary conditions for k-tuples of such operators to be Fredholm in the sense of Taylor and show they are sufficient in the case of the unit disk.

Correlation functions of the XXZ spin chain in the critical regime is studied at zerotemperature. They are exactly represented in the Fredholm determinant form and are related with an operator-valued Riemann–Hilbert problem. Analyzing this problem we prove that a two-point correlation function consisting of sufficiently separated spin operators is expressed by power-functions of the distance be...