In this introductory review, we study Hankel and Toeplitz opera- tors considering them as acting on certain spaces of analytic functions, namely Hardy compare their spectral properties such compactness criteria. contrast to operators, the symbol a operator is not uniquely determined by operator. We also connect operators with Fredholm give some most beautiful essential spectrum continuous index...

Basic results in the rigorous theory of weighted dynamical zeta functions or dynamically defined generalized Fredholm determinants are presented. Analytic properties of the zeta functions or determinants are related to statistical properties of the dynamics via spectral properties of dynamical transfer operators, acting on Banach spaces of observables.

Transfer operators and zeta functions of piecewise monotonie and of more general piecewise invertible dynamical systems are studied. To this end we construct Markov extensions of given systems, develop a kind of Fredholm theory for them, and carry the results back to the original systems. This yields e.g. bounds on the number of ergodic maximal measures or equilibrium states.

Space and time dependent correlation functions in the Heisenberg XX0 chain (in transverse magnetic field) are expressed in terms of Fredholm determinants of linear integral operators at all temperatures. The obtained expression allows useful computation of spectral shapes. Moreover, these determinant expressions allow to evaluate the asymptotic behaviour of correlation functions.

This article is about erroneous attempts to weaken the standard definition of unbounded Kasparov module (or spectral triple). This issue has been addressed previously, but here we present concrete counterexamples to claims in the literature that Fredholm modules can be obtained from these weaker variations of spectral triple. Our counterexamples are constructed using self-adjoint extensions of ...

In this paper, we prove the existence of mild solutions for the semilinear fractional order functional of Volterra-Fredholm type differential equations with impulses in a Banach space. The results are obtained by using the theory of fractional calculus, the analytic semigroup theory of linear operators and the fixed point techniques. ifx

We construct explicit generators of the K-theory and K-homology for the coordinate algebra of ‘functions’ on the quantum projective spaces. We also sketch a construction of unbounded Fredholm modules, that is to say Dirac-like operators and spectral triples of any positive real dimension.

In this article, we give some results concerning the continuity of the nonlinear Volterra and Fredholm integral operators on the space L1[0,∞). Then by using the concept of measure of weak noncompactness, we prove an existence result for a functional integral equation which includes several classes of nonlinear integral equations. Our results extend some previous works.

I will discuss index theory in the context of Poincare-Einstein manifolds of AdS/CFT fame. Complications arise because Dirac operators are not Fredholm and the Atiyah-Singer integrands are not integrable. In some cases, renormalization allows us to circumvent these difficulties.