In this paper we completely characterize compact Toeplitz operators on the harmonic Bergman space. By using this result we establish the short exact sequences associated with the Toeplitz algebra and the Hankel algebra. We show that the Fredholm index of each Fredholm operator in the Toeplitz algebra or the Hankel algebra is zero.  2002 Elsevier Science (USA). All rights reserved.

Recently, a beautiful identity due to Borodin and Okounkov was proved for Toeplitz determinants which shows how one can write a Toeplitz determinant as a Fredholm determinant. In this note we generalize this to the Wiener-Hopf case. The proof in the Wiener-Hopf case follows identically with the second one given in [1]. We include it here for completeness sake and because the nature of the ident...

Finite-dimensional perturbing operators are constructed using some incomplete information about eigen-solutions of an original and/or adjoint generalized Fredholm operator equation (with zero index). Adding such perturbing operator to the original one reduces the eigen-space dimension and can, particularly, lead to an unconditionally and uniquely solvable perturbed equation. For the second kind...

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...

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 ...

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...

The theory of measures of noncompactness has many applications on topology, functional analysis, and operator theory. In this paper, we consider one axiomatic approach to this notion which includes the most important classical definitions. We give some results concerning a certain class of semi-Fredholm and Fredholm operators via the concept of measures of noncompactness. Moreover, we establish...

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 ...

one can quantize an invertible symbol σ ∈ C∞(S∗X; hom(π∗E, π∗F )) as an elliptic pseudodifferential operator in the Φ-b or edge calculus, ΨΦ-b(X;E,F ), introduced in [7] or alternately as an elliptic operator in the Φ-c or φ calculus, ΨΦ-c(X;E,F ), introduced in [8]. As on a closed manifold, either of these operators will induce a bounded operator acting between natural L-spaces of sections but...