$K$-homology and Fredholm operators II: elliptic operators
نویسندگان
چکیده
منابع مشابه
K-Theory and Elliptic Operators
This expository paper is an introductory text on topologicalK-theory and the Atiyah-Singer index theorem, suitable for graduate students or advanced undegraduates already possessing a background in algebraic topology. The bulk of the material presented here is distilled from Atiyah’s classic “K-Theory” text, as well as his series of seminal papers “The Index of Elliptic Operators” with Singer. ...
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Let X be a compact metric space. By results of Brown, Douglas and Fillmore, [BDF2], the K-homology of X is realized by Ext(X), the equivalence classes of unital and essential extensions of C(X) by the compact operators K on a separable infinite dimensional Hilbert space H , or equivalently, the equivalence classes of unital and injective ∗-homomorphisms C(X) → Q, where Q = L(H)/K is the Calkin ...
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(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...
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The fundamental group iri of an arcwise connected topological space determines the one-dimensional homology groups of that space; indeed, the homology group Hi formed from finite chains with integral coefficients is simply the fundamental group made abelian. On the next dimension the second homotopy group 7r2 has a natural homomorphism 7r2—>Lf2 into the second integral homology group. The image...
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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...
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ژورنال
عنوان ژورنال: Pure and Applied Mathematics Quarterly
سال: 2016
ISSN: 1558-8599,1558-8602
DOI: 10.4310/pamq.2016.v12.n2.a2