Index of Fredholm operators

Fredholm Operators and the Generalized Index

One of the most fundamental problems in mathematics is to solve linear equations of the form Tf = g, where T is a linear transformation, g is known, and f is some unknown quantity. The simplest example of this comes from elementary linear algebra, which deals with solutions to matrix-vector equations of the form Ax = b. More generally, if V,W are vector spaces (or, in particular, Hilbert or Ban...

Fredholm Operators and the Family Index

The Fredholm index of a pair of commuting operators

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...

A Note on the Index of B-fredholm Operators

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...

Notes on Fredholm operators

(2) If K ∈ B(X) is compact, then for all λ ∈ C \ {0}, K − λ1 is Fredholm with index zero. (3) The shift operator S± ∈ B(`p) for 1 ≤ p ≤ ∞ defined by (S±x)n = xn±1 is Fredholm with index ±1. (4) If X,Y are finite dimensional and T ∈ B(X,Y ), then by the Rank-Nullity Theorem, ind(T ) = dim(X)− dim(Y ). Lemma 3. Suppose E,F ⊆ X are closed subspaces with F finite dimensional. (1) The subspace E + F...