Under minimal assumptions, we characterize the Fredholmity and compute the Fredholm index of abstract differential operators −d/dt + A(·) acting on spaces of functions f : R → X. Here A(t) are (in general) unbounded operators on the Banach space X and our results are formulated in terms of exponential dichotomies on two halflines for the propagtor solving the evolution equation u̇(t) = A(t)u(t) ...

This paper is concerned with the influence of frequency modulation on the semiFredholm properties of Toeplitz operators with oscillating matrix symbols. The main results give conditions on an orientation-preserving homeomorphism α of the real line that ensure the following: if b belongs to a certain class of oscillating matrix functions (periodic, almost periodic, or semi-almost periodic matrix...

From Corollary 3.5 in [Berkani, M; Sarih, M.; Studia Math. 148 (2001), 251– 257] we know that if S, T are commuting B-Fredholm operators acting on a Banach space X, then ST is a B-Fredholm operator. In this note we show that in general we do not have ind(ST ) = ind(S) + ind(T ), contrarily to what has been announced in Theorem 3.2 in [Berkani, M; Proc. Amer.Math. Soc. 130 (2002), 1717–1723]. Ho...

This paper concerns Fredholm theory in several variables, and its applications to Hilbert spaces of analytic functions. One feature is the introduction of ideas from commutative algebra to operator theory. Specifically, we introduce a method to calculate the Fredholm index of a pair of commuting operators. To achieve this, we define and study the Hilbert space analogs of Samuel multiplicities i...

Wiener-Hopf plus Hankel operators acting between Lebesgue spaces on the real line are studied in view of their invertibility, one sided-invertibility, Fredholm property, and the so-called n and d–normal properties. This is done in two different cases: (i) when the Fourier symbols of the operators are unitary functions, and (ii) when the Fourier symbols are related with sectorial elements appear...

The Fredholm properties of Toeplitz operators on the Bergman space A have been well-known for continuous symbols since the 1970s. We investigate the case p = 1 with continuous symbols under a mild additional condition, namely that of the logarithmic vanishing mean oscillation in the Bergman metric. Most differences are related to boundedness properties of Toeplitz operators acting on A that ari...

For smooth hyperbolic dynamical systems and smooth weights, we relate Ruelle transfer operators with dynamical Fredholm determinants and dynamical zeta functions: First, we establish bounds for the essential spectral radii of the transfer operator on new spaces of anisotropic distributions, improving our previous results [7]. Then we give a new proof of Kitaev’s [17] lower bound for the radius ...

This paper studies Dirac operators on end-periodic spin manifolds of dimension at least 4. We provide a necessary and sufficient condition for such an operator to be Fredholm for a generic end-periodic metric. We make use of end-periodic Dirac operators to give an analytical interpretation of an invariant of non-orientable smooth 4-manifolds due to Cappell and Shaneson. From this interpretation...

In this article, we consider Drazin invertible operators for study of the relationship between their B-Fredholm spectra and transfer some spectral properties defined through class operators. Among other results, investigate generalized a-Weyl’s theorem from T to inverse S, if it exists.

We investigate the relationship between the decay at infinity of the right-hand side f and solutions u of an equation Lu = f when L is a second order elliptic operator on RN . It is shown that when L is Fredholm, u inherits the type of decay of f (for instance, exponential, or power-like). In particular, the generalized eigenfunctions associated with all the Fredholm eigenvalues of L, isolated ...