We give some results concerning stability in the Fredholm operators and Browder operators set, via the concept of measure of noncompactness. Moreover, we prove some localization results on the essential spectra of bounded operators on Banach space. As application, we describe the essential spectra of weighted shift operators. Finally, we describe the spectra of polynomially compact operators, a...

Let H1 and H2 be Hilbert spaces and let B(H1,H2) be the space of bounded linear operators A : H1 → H2. An operator A ∈ B(H1,H2) is said to be Fredholm if first, the kernel ofA is finite-dimensional, and second the image ofA is closed and has finite codimension. An application of the open mapping theorem shows that the closedness requirement on the image is redundant. A well-known example of Fre...

Certain norm-related functions have been considered in order to obtain characterisations and perturbation results for various classes of semi-Fredholm-type operators. In the §2, by means of what we have called a perturbation function, we present a general approach to the question of obtaining characterisations and perturbation theorems for F + and strictly singular operators. To this end, for e...

For a continuous complex-valued function g on the real line without zeros, several notions of a mean winding number are introduced. We give necessary conditions for a Toeplitz operator with matrix-valued symbol G to be semi-Fredholm in terms of mean winding numbers of detG. The matrix function G is assumed to be continuous on the real line, and no other apriori assumptions on it are made.

Consider a selfadjoint unbounded operator D on a Hilbert space H and a one parameter norm continuous family of selfadjoint bounded operators {A(t) | t ∈ R} that converges in norm to asymptotes A± at ±∞. Then under certain conditions [RoSa95] that include the assumption that the operators {D(t) = D + A(t), t ∈ R} all have discrete spectrum then the spectral flow along the path {D(t)} can be show...

We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the heat operator in a general semi-finite von Neumann algebra. Our results have several applications. We deduce a formula for the Chern character of an odd L-summable Breuer-Fredholm module in terms of a Hochschild 1-cycle. We explain how to derive a Wodzicki res...