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href="/nlab/show/diagram+of+LCTVS+properties">locally convex topological vector spaces</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+convex+topological+vector+space">locally convex topological vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Banach+space">Banach Spaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/reflexive+Banach+space">reflexive</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Smith+space+%28functional+analysis%29">Smith Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert Spaces</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+space">Fréchet Spaces</a>, <a class="existingWikiWord" href="/nlab/show/Sobolev+space">Sobolev spaces</a>, <a class="existingWikiWord" href="/nlab/show/Lebesgue+space">Lebesgue Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bornological+vector+space">Bornological Vector Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/barrelled+topological+vector+space">Barrelled Vector Spaces</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+operator">linear operator</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/bounded+linear+operator">bounded</a>, <a class="existingWikiWord" href="/nlab/show/unbounded+linear+operator">unbounded</a>, <a class="existingWikiWord" href="/nlab/show/self-adjoint+operator">self-adjoint</a>, <a class="existingWikiWord" href="/nlab/show/compact+operator">compact</a>, <a class="existingWikiWord" href="/nlab/show/Fredholm+operator">Fredholm</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum+of+an+operator">spectrum of an operator</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebras">operator algebras</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/functional+calculus">functional calculus</a></li> </ul> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Stone-Weierstrass+theorem">Stone-Weierstrass theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+theory">spectral theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+theorem">spectral theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functional+calculus">functional calculus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Riesz+representation+theorem">Riesz representation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measure+theory">measure theory</a></p> </li> </ul> <h2 id="topics_in_functional_analysis">Topics in Functional Analysis</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/basis+in+functional+analysis">Bases</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theories+in+functional+analysis">Algebraic Theories in Functional Analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/an+elementary+treatment+of+Hilbert+spaces">An Elementary Treatment of Hilbert Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isomorphism+classes+of+Banach+spaces">When are two Banach spaces isomorphic?</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/functional+analysis+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="index_theory">Index theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/index+theory">index theory</a>, <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+topology">noncommutative topology</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a></p> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+stable+homotopy+theory">noncommutative stable homotopy theory</a></p> <p><strong><a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/genus">genus</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientation in generalized cohomology</a></strong></p> <h2 id="definitions">Definitions</h2> <p><strong><a class="existingWikiWord" href="/nlab/show/operator+K-theory">operator K-theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+module">Hilbert module</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/K-homology">K-homology</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Fredholm+operator">Fredholm operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+operator">differential operator</a>, <a class="existingWikiWord" href="/nlab/show/pseudodifferential+operator">pseudodifferential operator</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symbol+of+a+differential+operator">symbol of a differential operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+operator">elliptic operator</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+complex">elliptic complex</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Spin%5Ec+Dirac+operator">Spin^c Dirac operator</a></li> </ul> </li> </ul> <h2 id="index_theorems">Index theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+index">topological index</a>, <a class="existingWikiWord" href="/nlab/show/analytical+index">analytical index</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah-Singer+index+theorem">Atiyah-Singer index theorem</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Gauss-Bonnet+theorem">Gauss-Bonnet theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hirzebruch-Riemann-Roch+theorem">Hirzebruch-Riemann-Roch theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+analytic+index+theory">global analytic index theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hirzebruch+signature+theorem">Hirzebruch signature theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mishchenko-Fomenko+index+theorem">Mishchenko-Fomenko index theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baum-Connes+conjecture">Baum-Connes conjecture</a></p> </li> </ul> <h2 id="higher_genera">Higher genera</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+genus">elliptic genus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#generalizations'>Generalizations</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <div class="num_defn" id="FredholmOperator"> <h6 id="definition_2">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous</a> <a class="existingWikiWord" href="/nlab/show/linear+operator">linear operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><msub><mi>B</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>B</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">F \colon B_1\to B_2</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/Banach+spaces">Banach spaces</a> is <strong>Fredholm</strong> if it has <a class="existingWikiWord" href="/nlab/show/finite+set">finite</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> and finite dimensional <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a>.</p> </div> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>The difference between the <a class="existingWikiWord" href="/nlab/show/dimension+of+a+vector+space">dimensions</a> of the kernel and the cokernel of a Fredholm operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is called its <em><a class="existingWikiWord" href="/nlab/show/index">index</a></em> (the <em><a class="existingWikiWord" href="/nlab/show/Fredholm+index">Fredholm index</a></em>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ind</mi><mi>F</mi><mo>≔</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>ker</mi><mi>F</mi><mo stretchy="false">)</mo><mo>−</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>coker</mi><mi>F</mi><mo stretchy="false">)</mo><mo>=</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>ker</mi><mi>F</mi><mo stretchy="false">)</mo><mo>−</mo><mi>codim</mi><mo stretchy="false">(</mo><mi>im</mi><mi>F</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> ind F \coloneqq dim (ker F) - dim (coker F) = dim (ker F) - codim (im F) \,. </annotation></semantics></math></div></div> <p>A standard equivalent characterization of Fredholm operators is the following:</p> <div class="num_defn" id="Parametrix"> <h6 id="definition_4">Definition</h6> <p>A <strong>parametrix</strong> of a <a class="existingWikiWord" href="/nlab/show/bounded+operator">bounded</a> <a class="existingWikiWord" href="/nlab/show/linear+operator">linear operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><msub><mi>ℋ</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>ℋ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">F \colon \mathcal{H}_1 \to \mathcal{H}_2</annotation></semantics></math> is a reverse operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo lspace="verythinmathspace">:</mo><msub><mi>ℋ</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>ℋ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">P \colon \mathcal{H}_2 \to \mathcal{H}_1</annotation></semantics></math> which is an “<a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> up to <a class="existingWikiWord" href="/nlab/show/compact+operators">compact operators</a>”, i.e. such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∘</mo><mi>P</mi><mo>−</mo><msub><mi>id</mi> <mrow><msub><mi>ℋ</mi> <mn>2</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">F \circ P - id_{\mathcal{H}_2}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>∘</mo><mi>F</mi><mo>−</mo><msub><mi>id</mi> <mrow><msub><mi>ℋ</mi> <mn>1</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">P \circ F - id_{\mathcal{H}_1}</annotation></semantics></math> are both <a class="existingWikiWord" href="/nlab/show/compact+operators">compact operators</a>.</p> </div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/bounded+operator">bounded</a> <a class="existingWikiWord" href="/nlab/show/linear+operator">linear operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><msub><mi>B</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>B</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> F \colon B_1\to B_2</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/Banach+spaces">Banach spaces</a> is Fredholm, def. <a class="maruku-ref" href="#FredholmOperator"></a> precisely it is has a parametrix, def. <a class="maruku-ref" href="#Parametrix"></a>.</p> </div> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+operator">Elliptic operators</a> on <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a> are naturally Fredholm, when understood between the appropriate <a class="existingWikiWord" href="/nlab/show/Sobolev+spaces">Sobolev spaces</a>.</p> </li> <li> <p><a href="Dirac+field#FreeDiracFieldInCoulombBackground">charged vacua of free Dirac field in Coulomb background</a> are characterized by Fredholm operators</p> </li> </ul> <h2 id="properties">Properties</h2> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/image">image</a> (range) of a Fredholm operator is closed.</p> </div> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>The subspace <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fred</mi><mo stretchy="false">(</mo><msub><mi>B</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>B</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⊂</mo><mi>B</mi><mo stretchy="false">(</mo><msub><mi>B</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>B</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Fred(B_1,B_2)\subset B(B_1,B_2)</annotation></semantics></math> of Fredholm operators in the space of bounded linear operators with the norm topology is open.</p> </div> <p>In other words, given a Fredholm operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>, there exists <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\epsilon\gt 0</annotation></semantics></math> such that every bounded linear operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">‖</mo><mi>G</mi><mo>−</mo><mi>F</mi><mo stretchy="false">‖</mo><mo><</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\| G-F\|\lt \epsilon</annotation></semantics></math> is Fredholm. Fredholm operators on a fixed separable Hilbert space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>=</mo><msub><mi>B</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>B</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">H = B_1 = B_2</annotation></semantics></math> form a <a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a> with respect to the composition and the index is a morphism of semigroups: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ind</mi><mi>F</mi><mi>G</mi><mo>=</mo><mi>ind</mi><mi>F</mi><mo>+</mo><mi>ind</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">ind F G = ind F + ind G</annotation></semantics></math>.</p> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>The space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fred</mi></mrow><annotation encoding="application/x-tex">Fred</annotation></semantics></math> of all Fredholm operators on an (infinite dimensional) <a class="existingWikiWord" href="/nlab/show/separable+Hilbert+space">separable Hilbert space</a> is a model for the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> of degree-0 <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a>.</p> </div> <p>(…)</p> <h2 id="generalizations">Generalizations</h2> <p>Fredholm operators generalize to Fredholm complexes. A finite <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msub><mi>C</mi> <mn>0</mn></msub><mover><mo>→</mo><mrow><msub><mi>d</mi> <mn>0</mn></msub></mrow></mover><msub><mi>C</mi> <mn>1</mn></msub><mover><mo>→</mo><mrow><msub><mi>d</mi> <mn>1</mn></msub></mrow></mover><msub><mi>C</mi> <mn>2</mn></msub><mi>…</mi><msub><mi>C</mi> <mi>n</mi></msub><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to C_0 \stackrel{d_0}\to C_1\stackrel{d_1}\to C_2 \ldots C_n\to 0 </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>s and bounded operators is said to be a <strong>Fredholm complex</strong> if the images <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">d_i</annotation></semantics></math> are closed and the <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a> of the complex is finite dimensional. The <a class="existingWikiWord" href="/nlab/show/Euler+characteristic">Euler characteristic</a> (the alternative sum of the dimensions of the homology groups) is then called the <strong>index</strong> of the Fredholm complex. Each Fredholm operator can be considered as a Fredholm complex concentrated at zero. Each Fredholm complex produces a Fredholm operator from the sum of the even- to the sum of the odd-numbered spaces in the complex.</p> <p>One can consider <em>Fredholm almost complexes</em>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub><mo>∘</mo><msub><mi>d</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">d_i \circ d_{i-1}</annotation></semantics></math> is not zero but the image of that operator is compact. Out of every Fredholm almost complex one can make a complex by correcting the differentials by compact perturbation terms. <a class="existingWikiWord" href="/nlab/show/elliptic+complex">Elliptic complexes</a> give examples of Fredholm complexes.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Fredholm+module">Fredholm module</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fredholm+determinant">Fredholm determinant</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Toeplitz+operator">Toeplitz operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a>, <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/index+theory">index theory</a></p> </li> </ul> <h2 id="references">References</h2> <p>Textbook accounts:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/William+Arveson">William Arveson</a>, Section 3.3 of: <em>A Short Course on Spectral Theory</em>, Graduate Texts in Mathematics <strong>209</strong>, Springer (2002) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://link.springer.com/book/10.1007/b97227">doi:10.1007/b97227</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <p>Discussion of the space of Fredholm operators as the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> for <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a>:</p> <ul> <li id="Atiyah67"><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, Appendix of: <em>K-theory</em>, Harvard Lecture 1964 (notes by D. W. Anderson), Benjamin 1967 (<a href="https://www.maths.ed.ac.uk/~v1ranick/papers/atiyahk.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/AtiyahKTheory.pdf" title="pdf">pdf</a>)</li> </ul> <p>See also:</p> <ul> <li> <p>Wikipedia, <em><a href="http://en.wikipedia.org/wiki/Fredholm_operator">Fredholm operator</a></em></p> </li> <li> <p>A. S. Mishchenko, Векторные расслоения и их применения (Vector bundles and their applications), Nauka, Moscow, 1984. 208 pp. MR87f:55010</p> </li> <li> <p>S. Rempel, B-W. Schulze, <em>Index theory of elliptic boundary problems</em>, Akademie–Verlag, Berlin, 1982.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lars+H%C3%B6rmander">Lars Hörmander</a>, <em>The analysis of linear partial differential operators. III. Pseudo-differential operators</em>, 1994, reprinted 2007.</p> </li> <li> <p>Pietro Aiena, <em>Fredholm and local spectral theory, with applications to multipliers</em>, <a href="http://www.springer.com/mathematics/analysis/book/978-1-4020-1830-5">book page</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Otgonbayar+Uuye">Otgonbayar Uuye</a>, <em>A simple proof of the Fredholm Alternative</em>, <a href="http://arxiv.org/abs/1011.2933">arxiv/1011.2933</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Alexander Grothendieck</a>, <em>La théorie de Fredholm</em>, Bulletin de la Société Mathématique de France <strong>84</strong> (1956), p. 319-384, <a href="http://www.numdam.org/item?id=BSMF_1956__84__319_0">numdam</a></p> </li> <li> <p>Marina Prokhorova, <em>Spectral Sections</em>, <a href="https://arxiv.org/abs/2008.04672">arXiv:2008.04672</a>.</p> </li> <li> <p>Marina Prokhorova, <em>Spaces of unbounded Fredholm operators. I. Homotopy equivalences</em>, <a href="https://arxiv.org/abs/2110.14359">arXiv:2110.14359</a>.</p> </li> <li> <p>Marina Prokhorova, <em>The continuity properties of discrete-spectrum families of Fredholm operators</em>, <a href="https://arxiv.org/abs/2201.09869">arXiv:2201.09869</a>.</p> </li> <li> <p>Marina Prokhorova, <em>From graph to Riesz continuity</em>, <a href="https://arxiv.org/abs/2202.03337">arXiv:2202.03337</a>.</p> </li> </ul> <p>For Fredholm complexes, see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>Fredholm complexes</em>, Quarterly Journal of Mathematics 21:4 (1970), 385–402. <a href="http://dx.doi.org/10.1093/qmath/21.4.385">doi</a>.</li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/analysis">analysis</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on March 12, 2023 at 04:32:03. 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