The index of a pseudo B-Fredholm operator will be defined and generalize the usual operator. This concept used to extend some known results in Fredholm’s theory. Among other results, nullity, deficiency, ascent descent extended for pseudo-Fredholm

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...

Lecture Notes taken by Nilufer Koldan In this lecture I will describe one of the most significant achievements of the second half of the XX century – the Atiyah-Singer index theorem. I will also discuss some more recent developments in the area as well as some open problems. 1. The definition of the index 1.1. The finite-dimensional case. Let A : V 1 → V 2 be a linear map between finite dimensi...

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...

We prove that the operator G, the closure of the first-order differential operator −d/dt + D(t) on L2(R, X), is Fredholm if and only if the not well-posed equation u′(t) = D(t)u(t), t ∈ R, has exponential dichotomies on R+ and R− and the ranges of the dichotomy projections form a Fredholm pair; moreover, the index of this pair is equal to the Fredholm index of G. Here X is a Hilbert space, D(t)...

An operator in a Banach space is called upper (lower) semi-Browder if it is upper (lower) semi-Fredholm and has a finite ascent (descent). We extend this notion to n-tuples of commuting operators and show that this notion defines a joint spectrum. Further we study relations between semi-Browder and (essentially) semiregular operators. Denote by L(X) the algebra of all bounded linear operators i...

In the case of linear operator property P(B, k) was introduced by M.A. Kaashoek. this paper, we characterize essentially semi regular relation in terms k). After that and as an application result give some connection between relations. Further, will supplementary conditions on to be Fredholm. Then, analyze stability class relations under small perturbations Riesz operators. Finally, study prope...