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<title>Transfer operator</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Transfer operator</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Transfer_homomorphism" class="mw-redirect" title="Transfer homomorphism">transfer homomorphism</a>.</div>

<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, the <b>transfer operator</b> encodes information about an <a href="Iterated_map" class="mw-redirect" title="Iterated map">iterated map</a> and is frequently used to study the behavior of <a href="Dynamical_systems" class="mw-redirect" title="Dynamical systems">dynamical systems</a>, <a href="Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>, <a href="Quantum_chaos" title="Quantum chaos">quantum chaos</a> and <a href="Fractals" class="mw-redirect" title="Fractals">fractals</a>. In all usual cases, the largest eigenvalue is 1, and the corresponding eigenvector is the <a href="Invariant_measure" title="Invariant measure">invariant measure</a> of the system.

</p><p>The transfer operator is sometimes called the <b>Ruelle operator</b>, after <a href="David_Ruelle" title="David Ruelle">David Ruelle</a>, or the <b>Perron–Frobenius operator</b> or <b>Ruelle–Perron–Frobenius operator</b>, in reference to the applicability of the <a href="Perron%E2%80%93Frobenius_theorem" title="Perron–Frobenius theorem">Perron–Frobenius theorem</a> to the determination of the <a href="Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of the operator.

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>The iterated function to be studied is a map <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\rightarrow X}">

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<annotation encoding="application/x-tex">{\displaystyle f\colon X\rightarrow X}</annotation>

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</math></span><img src="./7fc989b42bdfd5a6cbf6f901340156fce82a8fc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.887ex; height:2.509ex;" alt="{\displaystyle f\colon X\rightarrow X}" loading="lazy"></span> for an arbitrary set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>.

</p><p>The transfer operator is defined as an operator <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation>

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</math></span><img src="./9027196ecb178d598958555ea01c43157d83597c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}" loading="lazy"></span> acting on the space of functions <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\Phi \colon X\rightarrow \mathbb {C} \}}">

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<annotation encoding="application/x-tex">{\displaystyle \{\Phi \colon X\rightarrow \mathbb {C} \}}</annotation>

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</math></span><img src="./3de0a03ed41b0bd1fd9216055389638b05bd8e5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.309ex; height:2.843ex;" alt="{\displaystyle \{\Phi \colon X\rightarrow \mathbb {C} \}}" loading="lazy"></span> as

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<annotation encoding="application/x-tex">{\displaystyle ({\mathcal {L}}\Phi )(x)=\sum _{y\,\in \,f^{-1}(x)}g(y)\Phi (y)}</annotation>

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</math></span><img src="./e66ddf1ee2112fba4ca15a44f29206f496df7dda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:28.15ex; height:6.343ex;" alt="{\displaystyle ({\mathcal {L}}\Phi )(x)=\sum _{y\,\in \,f^{-1}(x)}g(y)\Phi (y)}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon X\rightarrow \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle g\colon X\rightarrow \mathbb {C} }</annotation>

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</math></span><img src="./3f231b32aad4dc0b0895035a5997391868f5af1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.422ex; height:2.509ex;" alt="{\displaystyle g\colon X\rightarrow \mathbb {C} }" loading="lazy"></span> is an auxiliary valuation function. When <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle |J|}</annotation>

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</math></span><img src="./c428c92502a82b484670c344147db0e517316248.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.765ex; height:2.843ex;" alt="{\displaystyle |J|}" loading="lazy"></span>, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> is usually taken to be <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=1/|J|}">

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<annotation encoding="application/x-tex">{\displaystyle g=1/|J|}</annotation>

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</math></span><img src="./4f59526742f720653765fd5af9405f925aeb3edd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.304ex; height:2.843ex;" alt="{\displaystyle g=1/|J|}" loading="lazy"></span>.

</p><p>The above definition of the transfer operator can be shown to be the point-set limit of the measure-theoretic <a href="Pushforward_measure" title="Pushforward measure">pushforward</a> of <i>g</i>: in essence, the transfer operator is the <a href="Direct_image_functor" title="Direct image functor">direct image functor</a> in the category of <a href="Measurable_space" title="Measurable space">measurable spaces</a>. The left-adjoint of the Perron–Frobenius operator is the <a href="Koopman_operator" class="mw-redirect" title="Koopman operator">Koopman operator</a> or <a href="Composition_operator" title="Composition operator">composition operator</a>. The general setting is provided by the <a href="Borel_functional_calculus" title="Borel functional calculus">Borel functional calculus</a>.

</p><p>As a general rule, the transfer operator can usually be interpreted as a (left-)<a href="Shift_operator" title="Shift operator">shift operator</a> acting on a <a href="Shift_space" title="Shift space">shift space</a>. The most commonly studied shifts are the <a href="Subshifts_of_finite_type" class="mw-redirect" title="Subshifts of finite type">subshifts of finite type</a>. The adjoint to the transfer operator can likewise usually be interpreted as a right-shift. Particularly well studied right-shifts include the <a href="Jacobi_operator" title="Jacobi operator">Jacobi operator</a> and the <a href="Hessenberg_matrix" title="Hessenberg matrix">Hessenberg matrix</a>, both of which generate systems of <a href="Orthogonal_polynomials" title="Orthogonal polynomials">orthogonal polynomials</a> via a right-shift.

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<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<p>Whereas the iteration of a function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> naturally leads to a study of the orbits of points of X under iteration (the study of <a href="Chaos_theory" title="Chaos theory">point dynamics</a>), the transfer operator defines how (smooth) maps evolve under iteration. Thus, transfer operators typically appear in <a href="Physics" title="Physics">physics</a> problems, such as <a href="Quantum_chaos" title="Quantum chaos">quantum chaos</a> and <a href="Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>, where attention is focused on the time evolution of smooth functions. In turn, this has medical applications to <a href="Rational_drug_design" class="mw-redirect" title="Rational drug design">rational drug design</a>, through the field of <a href="Molecular_dynamics" title="Molecular dynamics">molecular dynamics</a>.

</p><p>It is often the case that the transfer operator is positive, has discrete positive real-valued <a href="Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a>, with the largest eigenvalue being equal to one. For this reason, the transfer operator is sometimes called the Frobenius–Perron operator.

</p><p>The <a href="Eigenfunction" title="Eigenfunction">eigenfunctions</a> of the transfer operator are usually fractals. When the logarithm of the transfer operator corresponds to a quantum <a href="Hamiltonian_(quantum_theory)" class="mw-redirect" title="Hamiltonian (quantum theory)">Hamiltonian</a>, the eigenvalues will typically be very closely spaced, and thus even a very narrow and carefully selected <a href="Quantum_ensemble" class="mw-redirect" title="Quantum ensemble">ensemble</a> of quantum states will encompass a large number of very different fractal eigenstates with non-zero <a href="Support_(mathematics)" title="Support (mathematics)">support</a> over the entire volume. This can be used to explain many results from classical statistical mechanics, including the irreversibility of time and the increase of <a href="Entropy" title="Entropy">entropy</a>.

</p><p>The transfer operator of the <a href="Bernoulli_map" class="mw-redirect" title="Bernoulli map">Bernoulli map</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b(x)=2x-\lfloor 2x\rfloor }">

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<annotation encoding="application/x-tex">{\displaystyle b(x)=2x-\lfloor 2x\rfloor }</annotation>

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</math></span><img src="./60776609db275a3c22d8efd286751c556ab8831e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.124ex; height:2.843ex;" alt="{\displaystyle b(x)=2x-\lfloor 2x\rfloor }" loading="lazy"></span> is exactly solvable and is a classic example of <a href="Chaos_theory" title="Chaos theory">deterministic chaos</a>; the discrete eigenvalues correspond to the <a href="Bernoulli_polynomials" title="Bernoulli polynomials">Bernoulli polynomials</a>. This operator also has a continuous spectrum consisting of the <a href="Hurwitz_zeta_function" title="Hurwitz zeta function">Hurwitz zeta function</a>.

</p><p>The transfer operator of the Gauss map <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(x)=1/x-\lfloor 1/x\rfloor }">

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<annotation encoding="application/x-tex">{\displaystyle h(x)=1/x-\lfloor 1/x\rfloor }</annotation>

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</math></span><img src="./458a607446bd08d0243da9e6da432a3756f8f782.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.791ex; height:2.843ex;" alt="{\displaystyle h(x)=1/x-\lfloor 1/x\rfloor }" loading="lazy"></span> is called the <a href="Gauss%E2%80%93Kuzmin%E2%80%93Wirsing_operator" title="Gauss–Kuzmin–Wirsing operator">Gauss–Kuzmin–Wirsing (GKW) operator</a>. The theory of the GKW dates back to a hypothesis by Gauss on <a href="Continued_fraction" title="Continued fraction">continued fractions</a> and is closely related to the <a href="Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>.

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<ul><li><a href="Bernoulli_scheme" title="Bernoulli scheme">Bernoulli scheme</a></li>

<li><a href="Shift_of_finite_type" class="mw-redirect" title="Shift of finite type">Shift of finite type</a></li>

<li><a href="Krein%E2%80%93Rutman_theorem" title="Krein–Rutman theorem">Krein–Rutman theorem</a></li>

<li><a href="Transfer-matrix_method_(statistical_mechanics)" title="Transfer-matrix method (statistical mechanics)">Transfer-matrix method</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFGaspard1992" class="citation journal cs1">Gaspard, Pierre (1992). "r-adic one dimensional maps and the Euler summation formula". <i>J. Phys. A: Math. Gen</i>. <b>25</b> (8): <span class="nowrap">L483 –</span> <span class="nowrap">L485</span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1992JPhA...25L.483G">1992JPhA...25L.483G</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0305-4470%2F25%2F8%2F017">10.1088/0305-4470/25/8/017</a>.</cite></li>

<li><cite id="CITEREFGaspard1998" class="citation book cs1">Gaspard, Pierre (1998). <i>Chaos, scattering and statistical mechanics</i>. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-39511-9</bdi>.</cite></li>

<li><cite id="CITEREFMackey1992" class="citation book cs1">Mackey, Michael C. (1992). <i>Time's Arrow : The origins of thermodynamic behaviour</i>. Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-94093-6</bdi>.</cite></li>

<li><cite id="CITEREFMayer1978" class="citation book cs1">Mayer, Dieter H. (1978). <i>The Ruelle-Araki transfer operator in classical statistical mechanics</i>. Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-09990-5</bdi>.</cite></li>

<li><cite id="CITEREFRuelle1978" class="citation book cs1">Ruelle, David (1978). <i>Thermodynamic formalism: the mathematical structures of classical equilibrium statistical mechanics</i>. Addison–Wesley, Reading. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-201-13504-3</bdi>.</cite></li>

<li><cite id="CITEREFRuelle2002" class="citation journal cs1">Ruelle, David (2002). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/notices/200208/fea-ruelle.pdf">"Dynamical Zeta Functions and Transfer Operators"</a> <span class="cs1-format">(PDF)</span>. <i>Notices of the AMS</i>. <b>49</b> (8): <span class="nowrap">887–</span>895.</cite> <i>(Provides an introductory survey).</i></li></ul>

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</style><div id="Functional_analysis_(topics_–_glossary)364" style="font-size:114%;margin:0 4em"><a href="Functional_analysis" title="Functional analysis">Functional analysis</a> (<a href="List_of_functional_analysis_topics" title="List of functional analysis topics">topics</a> – <a href="Glossary_of_functional_analysis" title="Glossary of functional analysis">glossary</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Spaces</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Banach_space" title="Banach space">Banach</a></li>

<li><a href="Besov_space" title="Besov space">Besov</a></li>

<li><a href="Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a></li>

<li><a href="Hilbert_space" title="Hilbert space">Hilbert</a></li>

<li><a href="H%C3%B6lder_space" class="mw-redirect" title="Hölder space">Hölder</a></li>

<li><a href="Nuclear_space" title="Nuclear space">Nuclear</a></li>

<li><a href="Orlicz_space" title="Orlicz space">Orlicz</a></li>

<li><a href="Schwartz_space" title="Schwartz space">Schwartz</a></li>

<li><a href="Sobolev_space" title="Sobolev space">Sobolev</a></li>

<li><a href="Topological_vector_space" title="Topological vector space">Topological vector</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Barrelled_space" title="Barrelled space">Barrelled</a></li>

<li><a href="Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li>

<li><a href="Dual_space" title="Dual space">Dual</a> (<a href="Dual_space#Algebraic_dual_space" title="Dual space">Algebraic</a> / <a href="Dual_space#Continuous_dual_space" title="Dual space">Topological</a>)</li>

<li><a href="Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex</a></li>

<li><a href="Reflexive_space" title="Reflexive space">Reflexive</a></li>

<li><a href="Separable_space" title="Separable space">Separable</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a></li>

<li><a href="Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li>

<li><a href="Closed_graph_theorem_(functional_analysis)" title="Closed graph theorem (functional analysis)">Closed graph</a></li>

<li><a href="Uniform_boundedness_principle" title="Uniform boundedness principle">Uniform boundedness principle</a></li>

<li><a href="Kakutani_fixed-point_theorem#Infinite-dimensional_generalizations" title="Kakutani fixed-point theorem">Kakutani fixed-point</a></li>

<li><a href="Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman</a></li>

<li><a href="Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark</a></li>

<li><a href="Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Operators</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adjoint_operator" class="mw-redirect" title="Adjoint operator">Adjoint</a></li>

<li><a href="Bounded_operator" title="Bounded operator">Bounded</a></li>

<li><a href="Compact_operator" title="Compact operator">Compact</a></li>

<li><a href="Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a></li>

<li><a href="Normal_operator" title="Normal operator">Normal</a></li>

<li><a href="Nuclear_operator" title="Nuclear operator">Nuclear</a></li>

<li><a href="Trace_class" title="Trace class">Trace class</a></li>

<li><a href="Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li>

<li><a href="Unbounded_operator" title="Unbounded operator">Unbounded</a></li>

<li><a href="Unitary_operator" title="Unitary operator">Unitary</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebras</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Banach_algebra" title="Banach algebra">Banach algebra</a></li>

<li><a href="C*-algebra" title="C*-algebra">C*-algebra</a></li>

<li><a href="Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">Spectrum of a C*-algebra</a></li>

<li><a href="Operator_algebra" title="Operator algebra">Operator algebra</a></li>

<li><a href="Group_algebra_of_a_locally_compact_group" title="Group algebra of a locally compact group">Group algebra of a locally compact group</a></li>

<li><a href="Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann algebra</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Open problems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Invariant_subspace_problem" title="Invariant subspace problem">Invariant subspace problem</a></li>

<li><a href="Mahler's_conjecture" class="mw-redirect" title="Mahler's conjecture">Mahler's conjecture</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hardy_space" title="Hardy space">Hardy space</a></li>

<li><a href="Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Spectral theory of ordinary differential equations</a></li>

<li><a href="Heat_kernel" title="Heat kernel">Heat kernel</a></li>

<li><a href="Index_theorem" class="mw-redirect" title="Index theorem">Index theorem</a></li>

<li><a href="Calculus_of_variations" title="Calculus of variations">Calculus of variations</a></li>

<li><a href="Functional_calculus" title="Functional calculus">Functional calculus</a></li>

<li><a href="Integral_linear_operator" title="Integral linear operator">Integral linear operator</a></li>

<li><a href="Jones_polynomial" title="Jones polynomial">Jones polynomial</a></li>

<li><a href="Topological_quantum_field_theory" title="Topological quantum field theory">Topological quantum field theory</a></li>

<li><a href="Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a></li>

<li><a href="Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></li>

<li><a href="Distribution_(mathematics)" title="Distribution (mathematics)">Distribution</a> (or <a href="Generalized_function" title="Generalized function">Generalized functions</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Advanced topics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Approximation_property" title="Approximation property">Approximation property</a></li>

<li><a href="Balanced_set" title="Balanced set">Balanced set</a></li>

<li><a href="Choquet_theory" title="Choquet theory">Choquet theory</a></li>

<li><a href="Weak_topology" title="Weak topology">Weak topology</a></li>

<li><a href="Banach%E2%80%93Mazur_distance" class="mw-redirect" title="Banach–Mazur distance">Banach–Mazur distance</a></li>

<li><a href="Tomita%E2%80%93Takesaki_theory" title="Tomita–Takesaki theory">Tomita–Takesaki theory</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</li></ul>

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This article is issued from <a class="external text" title="Last edited on 2025-01-06" href="https://en.wikipedia.org/wiki/?title=Transfer_operator&oldid=1267807566">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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