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In [[mathematics]], the '''Fredholm determinant''' is a [[complex-valued function]] which generalizes the [[determinant]] of a finite dimensional [[linear map|linear operator]]. It is defined for [[bounded operator]]s on a [[Hilbert space]] which differ from the [[identity operator]] by a [[trace-class operator]] (i.e. an operator whose singular values sum up to a finite number). The function is named after the [[mathematician]] [[Erik Ivar Fredholm]].

Fredholm determinants have had many applications in [[mathematical physics]], the most celebrated example being [[Gábor Szegő]]'s [[Szegő limit theorems|limit formula]], proved in response to a question raised by [[Lars Onsager]] and [[C. N. Yang]] on the [[spontaneous magnetization]] of the [[Ising model]].

==Definition==

=== Setup ===
Let <math>H</math> be a [[Hilbert space]] and <math>G</math> the set of [[bounded operator|bounded invertible operators]] on <math>H</math> of the form 
<math>I+T</math>, where <math>T</math> is a [[trace-class operator]]. <math>G</math> is a [[group (mathematics)|group]] because

* The set of trace-class operators is an ideal in the algebra of bounded linear operators, so <math>(I+T)(I+T')-I = T + T' + TT'</math> is trace-class.
* <math display="inline"> (I+T)^{-1} - I = - T(I+T)^{-1},</math> so <math>(I+T)^{-1}-I</math> is trace class if <math>T</math> is.

<math>G</math> has a natural [[Metric (mathematics)|metric]] given by <math>d(X,Y)=\|X-Y\|_1</math>, where <math>\|X\|_1 = \sum_i |\lambda_i(X)|</math> is the trace-class norm.

=== Definition by exponential trace ===
One definition uses the exponential trace formula. For finite-dimensional matrices, we have <math display="inline"> \det (I+A) = e^{\operatorname{Tr}(\ln (I+A))} </math>, which expands in Taylor series to<math display="block"> \operatorname{det}(I+A)=\exp \left(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \operatorname{Tr}\left(A^n\right)\right)</math>This then generalizes directly to trace-class operators.

=== Definition by exterior powers ===
[[File:N vector positive.svg|thumb|The exterior product of up to 3 vectors.]]
In the finite-dimensional case, the determinant of an operator can be interpreted as the factor by which it scales the (oriented) volume of a [[parallelepiped]]. This can be generalized to infinite dimensions.

In finite dimensions, by expanding the definition of determinant as a sum over permutations,<math display="block"> \det (I+ A) = \sum_{S} \det(A_{SS}) </math>where <math> S </math> ranges over all subsets of the index set of <math> A </math>. For example, when the index set is <math> \{1,2\} </math> then <math> S = \{\}, \{1\}, \{2\}, \{1,2\} </math>.

If <math>H</math> is an <math>n</math>-dimensional Hilbert space with [[inner product]] <math>(\cdot, \cdot)</math>, then the <math>k</math>-th [[exterior algebra|exterior power]] <math>\Lambda^k H</math> is also a <math>\binom{n}{k}</math>-dimensional Hilbert space, with inner product
<math display="block"> (v_1 \wedge v_2 \wedge \cdots \wedge v_k, w_1 \wedge w_2 \wedge \cdots \wedge w_k) = \det (v_i,w_j). </math>In particular
<math display="block"> e_{i_1} \wedge e_{i_2} \wedge \cdots \wedge e_{i_k}, \qquad (i_1<i_2<\cdots<i_k)</math>gives an [[orthonormal basis]] of <math>\Lambda^k H</math> if <math>(e_i)</math> is an orthonormal basis of <math>H</math>.

If <math>A</math> is an operator on <math>H</math>, then <math>A</math> [[functorial]]ly defines a bounded operator <math>\Lambda^k(A)</math> on <math>\Lambda^k H</math> by
<math display="block"> \Lambda^k(A) v_1 \wedge v_2 \wedge \cdots \wedge v_k = Av_1 \wedge Av_2 \wedge \cdots \wedge Av_k.</math>By definition of trace, we have<math display="block">\operatorname{Tr}\left(\Lambda^k A\right) = 
\sum_{1 \leq i_1<\cdots<i_k \leq n} (e_{i_1} \wedge e_{i_2} \wedge \cdots \wedge e_{i_k}, 
Ae_{i_1} \wedge Ae_{i_2} \wedge \cdots \wedge Ae_{i_k})</math>The summand simplifies to <math>\det[(e_{i_j}, A e_{i_{j'}})] = \det(A_{SS})</math> where <math>S = \{i_1, \dots, i_k\}</math>. Thus <math display="block">\begin{aligned}
&\operatorname{Tr} \Lambda^k(A)=\sum_{|S|=k} \operatorname{det}\left(A_{S S}\right) .\\
&\operatorname{det}(I+A)=\sum_{k=0}^n \operatorname{Tr} \Lambda^k(A) .
\end{aligned}</math>This generalizes to infinite-dimensional Hilbert spaces, and bounded trace-class operators, allowing us to define the '''Fredholm determinant''' by<math display="block"> \det (I+ A) = \sum_{k=0}^\infty \operatorname{Tr} \Lambda^k(A) </math>To show that the definition makes sense, note that if <math>A</math> is trace-class, then <math>\Lambda^k(A)</math> is also trace-class with <math display="inline"> \|\Lambda^k(A)\|_1 \le \|A\|_1^k/k!</math>, thus <math>\sum_{k=0}^\infty |\operatorname{Tr} \Lambda^k(A) | \leq e^{\|A\|_1}</math>.

{{hidden begin|style=width:100%|ta1=center|border=1px #aaa solid|title=Proof}}

{{Math proof|title=Proof|proof= We have <math display="inline">\|A\|_1=\sum_i \sigma_i</math> where <math display="inline">\sigma_i</math> are the singular values of <math display="inline">A</math>.

The singular values of <math display="inline">\Lambda^k(A)</math> are exactly the products of <math display="inline">k</math> distinct singular values of <math display="inline">A</math>. In other words, if you list all <math display="inline">k</math>-tuples with <math display="inline">1 \leq i_1<i_2<\cdots<i_k</math>, then the corresponding singular value of <math display="inline">\Lambda^k(A)</math> is <math display="inline">
\sigma_{i_1} \sigma_{i_2} \cdots \sigma_{i_k}
</math>

Thus,

<math display="block">
\left\|\Lambda^k(A)\right\|_1=\sum_{1 \leq i_1<i_2<\cdots<i_k} \sigma_{i_1} \sigma_{i_2} \cdots \sigma_{i_k}
</math>

This is the <math display="inline">k</math> th elementary symmetric function of the singular values of <math display="inline">A</math>. Let <math display="inline">a_i \geq 0</math> (in our case <math display="inline">a_i=\sigma_i</math> ) then by expanding the right side, we have

<math display="block">
\sum_{1 \leq i_1<\cdots<i_k} a_{i_1} \cdots a_{i_k} \leq \frac{1}{k!}\left(\sum_i a_i\right)^k
</math>

}}{{hidden end}}

==Properties==
By default, all operators are assumed trace-class.
* <math display="inline"> \det(I+A) \cdot \det(I+B) = \det(I+A)(I+B). </math>
* <math display="inline"> z \mapsto \det (I+ zA) = \sum_{k=0}^\infty z^k\operatorname{Tr} \Lambda^k(A)</math> defines an [[entire function]], with <math display="inline"> \left|\det (I+ zA)\right| \le \exp (|z|\cdot \|A\|_1). </math> 

* The function <math>A \mapsto \det(I+A)</math> is continuous on trace-class operators, with
<math display="block"> \left|\det(I+A) - \det(I+B)\right| \le \|A-B\|_1 \exp (\|A\|_1 + \|B\|_1 +1).</math> 
One can improve this inequality slightly to the following, as noted in {{Harvard citation|Simon|2005|loc=Chapter 5}}: 
<math display="block"> \left|\det(I+A) -\det(I+B)\right| \le \|A-B\|_1 \exp (\max(\|A\|_1,\|B\|_1) +1).</math>

* The function <math>\det</math> defines a [[homomorphism]] of type <math>G \to \mathbb{C}^\times </math> where <math>\mathbb C^\times</math> the multiplicative group of nonzero complex numbers (since elements of <math>G</math> are invertible).
* If <math>T</math> is in <math>G</math> and <math>X</math> is invertible, <math display="inline"> \det XTX^{-1} = \det T.</math>

* <math display="inline"> \det e^A = \exp \, \operatorname{Tr} (A).</math>

* <math display="inline"> \log \det (I+ zA) = \operatorname{Tr} (\log{(I+zA)}) = \sum_{k=1}^\infty (-1)^{k+1}\frac{\operatorname{Tr} A^k}{k}z^k</math>

==Integral operators ==
The Fredholm determinant is often applied to integral operators. Let the trace-class operator <math>T</math> be an [[integral operator]] given by a kernel <math>K(x,y)</math>, then the Fredholm determinant is defined, like before, by<math display="block">\det(I-\lambda T) 
= \sum_{n=0}^\infty (-\lambda)^n \operatorname{Tr } \Lambda^n(T) 
= \exp{\left(-\sum_{n=1}^\infty\frac{\operatorname{Tr}(T^n)}{n}\lambda^n\right)}</math>where <math>T</math> is an [[integral operator]]. The trace of the operator <math>T</math> and its alternating powers is given in terms of the kernel <math>K</math> by
<math display="block">\operatorname{Tr } T = \int K(x,x)\,dx</math>
and
<math display="block">\operatorname{Tr }\Lambda^2(T) = \frac{1}{2!}  \iint \left(K(x,x)K(y,y)-K(x,y) K(y,x)\right)dx \, dy</math>
and in general
<math display="block">\operatorname{Tr } \Lambda^n(T) = \frac{1}{n!}\int\cdots\int \det [K(x_i,x_j)]_{i, j \in 1:n}\,dx_{1:n}</math>The trace is well-defined for these kernels, since these are [[trace-class]] or [[nuclear operator]]s.

To see that this is a special case of the previous section's general definition, note that,<math display="block">\operatorname{Tr}\left(\Lambda^k A\right) = 
\sum_{1 \leq i_1<\cdots<i_k \leq n} (e_{i_1} \wedge e_{i_2} \wedge \cdots \wedge e_{i_k}, 
Ae_{i_1} \wedge Ae_{i_2} \wedge \cdots \wedge Ae_{i_k})</math>is equivalent to<math display="block">
\frac{1}{k!}\sum_{i_1, \cdots, i_k \in 1:n, \text{ all different}} \det(A_{SS})</math>where <math>S</math> is the ordered sequence <math>i_1, \dots, i_k</math>. Now, to convert this to integral equations, a matrix becomes a kernel, and a summation over indices becomes an integral over coordinates.

The above argument is intuitive. A proper definition requires a presentation showing that each of the manipulations are well-defined, convergent, and so on, for the given situation for which the Fredholm determinant is contemplated.  Since the kernel <math>K</math> may be defined for a large variety of [[Hilbert space]]s and [[Banach space]]s, this is a non-trivial exercise.

=== Integral equation ===
The original {{Harvard citation|Fredholm|1903}} considered the integral equation<math display="block">u(x)+z \int_a^b K(x, y) u(y) d y=f(x) \quad(x \in(a, b))</math>which can be written as <math>(I + zA)  u = f</math>. Fredholm proved that this equation has a unique solution iff <math>\det(I + zA) \neq 0</math>.

==Commutators==
A function <math>F(t)</math> from <math>(a,b)</math> into <math>G</math> is said to be ''differentiable'' if <math>F(t)-I</math> is differentiable as a map into the trace-class operators, i.e. if the limit

<math display="block"> \dot{F}(t) = \lim_{h \to 0} {F(t+h) - F(t)\over h}</math>

exists in trace-class norm.

If <math>g(t)</math> is a differentiable function with values in trace-class operators, then so too is <math>\exp g(t)</math> and

<math display="block"> F^{-1} \dot{F} = {\operatorname{id} -  \exp - \operatorname{ad} g(t)\over \operatorname{ad} g(t)} \cdot \dot{g}(t),</math>

where <math display="block"> \operatorname{ad}(X)\cdot Y = XY -YX.</math>

[[Israel Gohberg]] and [[Mark Krein]] proved that if <math>F</math> is a differentiable function into <math>G</math>, then <math>f=\det F</math> is a differentiable map into
<math>\mathbb{C}^*</math> with
<math display="block"> f^{-1} \dot{f} = \operatorname{Tr} F^{-1} \dot{F}. </math>

This result was used by Joel Pincus, William Helton and [[Roger Evans Howe|Roger Howe]] to prove that if <math>A</math> and <math>B</math> are bounded operators with trace-class commutator <math>AB-BA</math>, then

<math display="block"> \det e^A e^B e^{-A} e^{-B} = \exp \operatorname{Tr} (AB-BA). </math>

==Szegő limit formula==
{{See also |Szegő limit theorems}}

Let <math>H=L^2(S^1)</math> and let <math>P</math> be the [[orthogonal projection]] onto the [[Hardy space]] <math>H^2(S^1)</math>.

If <math>f</math> is a [[smooth function]] on the circle, let <math>m(f)</math> denote the corresponding [[multiplication operator]] on <math>H</math>.

The commutator
<math display="block"> Pm(f)-m(f)P</math>
is trace-class.

Let <math>T(f)</math> be the [[Toeplitz operator]] on <math>H^2(S^1)</math> defined by
<math display="block"> T(f) = Pm(f)P, </math>

then the additive commutator
<math display="block"> T(f) T(g) - T(g) T(f) </math>
is trace-class if <math>f</math> and <math>g</math> are smooth.

Berger and Shaw proved that
<math display="block"> \operatorname{tr}(T(f) T(g) - T(g) T(f)) =  {1\over 2\pi i} \int_0^{2\pi} f \, dg.</math>

If <math>f</math> and <math>g</math> are smooth, then
<math display="block"> T(e^{f+g})T(e^{-f}) T(e^{-g}) </math>
is in <math>G</math>.

[[Harold Widom]] used the result of Pincus-Helton-Howe to prove that
<math display="block"> \det T(e^f) T(e^{-f}) = \exp \sum_{ n>0} na_n a_{-n},</math>
where 
<math display="block"> f(z) =\sum a_n z^n.</math>

He used this to give a new proof of [[Gábor Szegő]]'s celebrated limit formula:
<math display="block"> \lim_{N\to \infty} \det P_N m(e^f) P_N = \exp \sum_{ n>0} na_n a_{-n},</math>
where <math>P_N</math> is the projection onto the subspace of <math>H</math> spanned by <math>1,z,\ldots,z^N</math> and <math>a_0=0</math>.

Szegő's limit formula was proved in 1951 in response to a question raised by the work [[Lars Onsager]] and [[C. N. Yang]] on the calculation of the [[spontaneous magnetization]] for the [[Ising model]]. The formula of Widom, which leads quite quickly to Szegő's limit formula, is also equivalent to the duality between [[boson]]s and [[fermion]]s in [[conformal field theory]]. A singular version of Szegő's limit formula for functions supported on an arc of the circle was proved by Widom; it has been applied to establish probabilistic results on the eigenvalue distribution of [[random matrix|random unitary matrices]].

==History==
The Fredholm determinant was first used in {{Harvard citation|Fredholm|1903}} to solve an integral equation. Realizing the potential, Hilbert wrote 6 papers during 1904 to 1910 (collected in {{Harvard citation|Hilbert|1924}}), beginning the theory of compact operators on Hilbert spaces. See {{Harvard citation|Bornemann|2010}} and references therein.

The Fredholm determinant was used by physicist [[John Archibald Wheeler|John A. Wheeler]] (1937, Phys. Rev. 52:1107) to help provide mathematical description of the wavefunction for a composite nucleus composed of antisymmetrized combination of partial wavefunctions by the method of Resonating Group Structure.  This method corresponds to the various possible ways of distributing the energy of neutrons and protons into fundamental boson and fermion nucleon cluster groups or building blocks such as the alpha-particle, helium-3, deuterium, triton, di-neutron, etc. When applied to the method of Resonating Group Structure for beta and alpha stable isotopes, use of the Fredholm determinant: (1) determines the energy values of the composite system, and (2) determines scattering and disintegration cross sections. The method of Resonating Group Structure of Wheeler provides the theoretical bases for all subsequent Nucleon Cluster Models and associated cluster energy dynamics for all light and heavy mass isotopes (see review of Cluster Models in physics in N.D. Cook, 2006).

==References==
* {{cite journal |last=Fredholm |first=Ivar |date=1903 |title=Sur une classe d’équations fonctionnelles |url=https://projecteuclid.org/journals/acta-mathematica/volume-27/issue-none/Sur-une-classe-d%c3%a9quations-fonctionnelles/10.1007/BF02421317.pdf |journal=Acta Mathematica |volume=27 |issue=0 |pages=365–390 |doi=10.1007/BF02421317 |issn=0001-5962 |access-date=February 7, 2025 |doi-access=free}}
* {{cite book |last=Hilbert |first=D. |url=https://books.google.com/books?id=ZsXPAAAAMAAJ&pg=PP3 |title=Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen |publisher=B. G. Teubner |year=1924 |series=Fortschritte der mathematischen Wissenschaften in Monographien |language=de}}
* {{cite book |last=Gohberg |first=Israel |title=Traces and Determinants of Linear Operators |last2=Goldberg |first2=Seymour |last3=Krupnik |first3=Nahum |date=2000 |publisher=Birkhäuser Basel |isbn=978-3-0348-9551-4 |publication-place=Basel |doi=10.1007/978-3-0348-8401-3 |doi-access=free}}
*{{citation| last=Simon|first=Barry| title=Trace Ideals and Their Applications|series=Mathematical Surveys and Monographs| volume=120| publisher=American Mathematical Society|year=2005| isbn=0-8218-3581-5}}
*{{cite journal | last=Wheeler | first=John A. | title=On the Mathematical Description of Light Nuclei by the Method of Resonating Group Structure | journal=Physical Review | publisher=American Physical Society (APS) | volume=52 | issue=11 | date=1937-12-01 | issn=0031-899X | doi=10.1103/physrev.52.1107 | pages=1107–1122| bibcode=1937PhRv...52.1107W }}
*{{citation|last=Bornemann|first=Folkmar|title=On the numerical evaluation of Fredholm determinants|journal= Math. Comp.| volume=79|pages=871–915|publisher=Springer|year=2010|issue=270 |doi=10.1090/s0025-5718-09-02280-7|arxiv=0804.2543}}

{{Hilbert space}}
{{Topological tensor products and nuclear spaces}}
{{Functional analysis}}

[[Category:Determinants]]
[[Category:Fredholm theory]]
[[Category:Hilbert spaces]]
[[Category:Topological tensor products]]