Fredholm determinant#Szegő limit formula

Let $H$ be a Hilbert space and $G$ the set of bounded invertible operators on $H$ of the form

$I+T$, where $T$ is a trace-class operator. $G$ is a group because

• The set of trace-class operators is an ideal in the algebra of bounded linear operators, so $(I+T)(I+T\text{'})-I\; =\; T\; +\; T\text{'}\; +\; TT\text{'}$ is trace-class.
• $(I+T)^\{-1\}\; -\; I\; =\; -\; T(I+T)^\{-1\},$ so $(I+T)^\{-1\}-I$ is trace class if $T$ is.

$G$ has a natural metric given by $d(X,Y)=\backslash |X-Y\backslash |\_1$, where $\backslash |X\backslash |\_1\; =\; \backslash sum\_i\; |\backslash lambda\_i(X)|$ is the trace-class norm.

One definition uses the exponential trace formula. For finite-dimensional matrices, we have $\backslash det\; (I+A)\; =\; e^\{\backslash operatorname\{Tr\}(\backslash ln\; (I+A))\}$, which expands in Taylor series to$$\backslash operatorname\{det\}(I+A)=\backslash exp\; \backslash left(\backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{(-1)^\{n+1\}\}\{n\}\; \backslash operatorname\{Tr\}\backslash left(A^n\backslash right)\backslash right)$$This then generalizes directly to trace-class operators.

File:N vector positive.svg

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,$$\backslash det\; (I+\; A)\; =\; \backslash sum\_\{S\}\; \backslash det(A\_\{SS\})$$where $S$ ranges over all subsets of the index set of $A$. For example, when the index set is $\backslash \{1,2\backslash \}$ then $S\; =\; \backslash \{\backslash \},\; \backslash \{1\backslash \},\; \backslash \{2\backslash \},\; \backslash \{1,2\backslash \}$.

If $H$ is an $n$-dimensional Hilbert space with inner product $(\backslash cdot,\; \backslash cdot)$, then the $k$-th exterior power $\backslash Lambda^k\; H$ is also a $\backslash binom\{n\}\{k\}$-dimensional Hilbert space, with inner product

$$(v\_1\; \backslash wedge\; v\_2\; \backslash wedge\; \backslash cdots\; \backslash wedge\; v\_k,\; w\_1\; \backslash wedge\; w\_2\; \backslash wedge\; \backslash cdots\; \backslash wedge\; w\_k)\; =\; \backslash det\; (v\_i,w\_j).$$In particular

If $A$ is an operator on $H$, then $A$ functorially defines a bounded operator $\backslash Lambda^k(A)$ on $\backslash Lambda^k\; H$ by

$$\backslash Lambda^k(A)\; v\_1\; \backslash wedge\; v\_2\; \backslash wedge\; \backslash cdots\; \backslash wedge\; v\_k\; =\; Av\_1\; \backslash wedge\; Av\_2\; \backslash wedge\; \backslash cdots\; \backslash wedge\; Av\_k.$$By definition of trace, we have$$\backslash operatorname\{Tr\}\backslash left(\backslash Lambda^k\; A\backslash right)\; =$$

\sum_{1 \leq i_1<\cdots

Ae_{i_1} \wedge Ae_{i_2} \wedge \cdots \wedge Ae_{i_k})The summand simplifies to $\backslash det[(e\_\{i\_j\},\; A\; e\_\{i\_\{j\text{'}\}\})]\; =\; \backslash det(A\_\{SS\})$ where $S\; =\; \backslash \{i\_1,\; \backslash dots,\; i\_k\backslash \}$. Thus $$\backslash begin\{aligned\}$$

&\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}This generalizes to infinite-dimensional Hilbert spaces, and bounded trace-class operators, allowing us to define the Fredholm determinant by$$\backslash det\; (I+\; A)\; =\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash operatorname\{Tr\}\; \backslash Lambda^k(A)$$To show that the definition makes sense, note that if $A$ is trace-class, then $\backslash Lambda^k(A)$ is also trace-class with $\backslash |\backslash Lambda^k(A)\backslash |\_1\; \backslash le\; \backslash |A\backslash |\_1^k/k!$, thus $\backslash sum\_\{k=0\}^\backslash infty\; |\backslash operatorname\{Tr\}\; \backslash Lambda^k(A)\; |\; \backslash leq\; e^\{\backslash |A\backslash |\_1\}$.

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

{{Math proof|title=Proof|proof= We have $\backslash |A\backslash |\_1=\backslash sum\_i\; \backslash sigma\_i$ where $\backslash sigma\_i$ are the singular values of $A$.

The singular values of $\backslash Lambda^k(A)$ are exactly the products of $k$ distinct singular values of $A$. In other words, if you list all $k$-tuples with

\sigma_{i_1} \sigma_{i_2} \cdots \sigma_{i_k}

Thus,

\left\|\Lambda^k(A)\right\|_1=\sum_{1 \leq i_1

This is the $k$ th elementary symmetric function of the singular values of $A$. Let $a\_i\; \backslash geq\; 0$ (in our case $a\_i=\backslash sigma\_i$ ) then by expanding the right side, we have

\sum_{1 \leq i_1<\cdots

}}{{hidden end}}

By default, all operators are assumed trace-class.

• $\backslash det(I+A)\; \backslash cdot\; \backslash det(I+B)\; =\; \backslash det(I+A)(I+B).$
• $z\; \backslash mapsto\; \backslash det\; (I+\; zA)\; =\; \backslash sum\_\{k=0\}^\backslash infty\; z^k\backslash operatorname\{Tr\}\; \backslash Lambda^k(A)$ defines an entire function, with $\backslash left|\backslash det\; (I+\; zA)\backslash right|\; \backslash le\; \backslash exp\; (|z|\backslash cdot\; \backslash |A\backslash |\_1).$

• The function $A\; \backslash mapsto\; \backslash det(I+A)$ is continuous on trace-class operators, with

$$\backslash left|\backslash det(I+A)\; -\; \backslash det(I+B)\backslash right|\; \backslash le\; \backslash |A-B\backslash |\_1\; \backslash exp\; (\backslash |A\backslash |\_1\; +\; \backslash |B\backslash |\_1\; +1).$$

One can improve this inequality slightly to the following, as noted in {{Harvard citation|Simon|2005|loc=Chapter 5}}:

$$\backslash left|\backslash det(I+A)\; -\backslash det(I+B)\backslash right|\; \backslash le\; \backslash |A-B\backslash |\_1\; \backslash exp\; (\backslash max(\backslash |A\backslash |\_1,\backslash |B\backslash |\_1)\; +1).$$

• The function $\backslash det$ defines a homomorphism of type $G\; \backslash to\; \backslash mathbb\{C\}^\backslash times$ where $\backslash mathbb\; C^\backslash times$ the multiplicative group of nonzero complex numbers (since elements of $G$ are invertible).
• If $T$ is in $G$ and $X$ is invertible, $\backslash det\; XTX^\{-1\}\; =\; \backslash det\; T.$

• $\backslash det\; e^A\; =\; \backslash exp\; \backslash ,\; \backslash operatorname\{Tr\}\; (A).$

• $\backslash log\; \backslash det\; (I+\; zA)\; =\; \backslash operatorname\{Tr\}\; (\backslash log\{(I+zA)\})\; =\; \backslash sum\_\{k=1\}^\backslash infty\; (-1)^\{k+1\}\backslash frac\{\backslash operatorname\{Tr\}\; A^k\}\{k\}z^k$

The Fredholm determinant is often applied to integral operators. Let the trace-class operator $T$ be an integral operator given by a kernel $K(x,y)$, then the Fredholm determinant is defined, like before, by$$\backslash det(I-\backslash lambda\; T)$$

= \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)}where $T$ is an integral operator. The trace of the operator $T$ and its alternating powers is given in terms of the kernel $K$ by

$$\backslash operatorname\{Tr\; \}\; T\; =\; \backslash int\; K(x,x)\backslash ,dx$$

and

$$\backslash operatorname\{Tr\; \}\backslash Lambda^2(T)\; =\; \backslash frac\{1\}\{2!\}\; \backslash iint\; \backslash left(K(x,x)K(y,y)-K(x,y)\; K(y,x)\backslash right)dx\; \backslash ,\; dy$$

and in general

$$\backslash operatorname\{Tr\; \}\; \backslash Lambda^n(T)\; =\; \backslash frac\{1\}\{n!\}\backslash int\backslash cdots\backslash int\; \backslash det\; [K(x\_i,x\_j)]\_\{i,\; j\; \backslash in\; 1:n\}\backslash ,dx\_\{1:n\}$$The trace is well-defined for these kernels, since these are trace-class or nuclear operators.

To see that this is a special case of the previous section's general definition, note that,$$\backslash operatorname\{Tr\}\backslash left(\backslash Lambda^k\; A\backslash right)\; =$$

\sum_{1 \leq i_1<\cdots

Ae_{i_1} \wedge Ae_{i_2} \wedge \cdots \wedge Ae_{i_k})is equivalent to$$$$

\frac{1}{k!}\sum_{i_1, \cdots, i_k \in 1:n, \text{ all different}} \det(A_{SS})where $S$ is the ordered sequence $i\_1,\; \backslash dots,\; i\_k$. 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 $K$ may be defined for a large variety of Hilbert spaces and Banach spaces, this is a non-trivial exercise.

The original {{Harvard citation|Fredholm|1903}} considered the integral equation$$u(x)+z\; \backslash int\_a^b\; K(x,\; y)\; u(y)\; d\; y=f(x)\; \backslash quad(x\; \backslash in(a,\; b))$$which can be written as $(I\; +\; zA)\; u\; =\; f$. Fredholm proved that this equation has a unique solution iff $\backslash det(I\; +\; zA)\; \backslash neq\; 0$.

A function $F(t)$ from $(a,b)$ into $G$ is said to be differentiable if $F(t)-I$ is differentiable as a map into the trace-class operators, i.e. if the limit

$$\backslash dot\{F\}(t)\; =\; \backslash lim\_\{h\; \backslash to\; 0\}\; \{F(t+h)\; -\; F(t)\backslash over\; h\}$$

exists in trace-class norm.

If $g(t)$ is a differentiable function with values in trace-class operators, then so too is $\backslash exp\; g(t)$ and

$$F^\{-1\}\; \backslash dot\{F\}\; =\; \{\backslash operatorname\{id\}\; -\; \backslash exp\; -\; \backslash operatorname\{ad\}\; g(t)\backslash over\; \backslash operatorname\{ad\}\; g(t)\}\; \backslash cdot\; \backslash dot\{g\}(t),$$

where $$\backslash operatorname\{ad\}(X)\backslash cdot\; Y\; =\; XY\; -YX.$$

Israel Gohberg and Mark Krein proved that if $F$ is a differentiable function into $G$, then $f=\backslash det\; F$ is a differentiable map into

$\backslash mathbb\{C\}^*$ with

$$f^\{-1\}\; \backslash dot\{f\}\; =\; \backslash operatorname\{Tr\}\; F^\{-1\}\; \backslash dot\{F\}.$$

This result was used by Joel Pincus, William Helton and Roger Howe to prove that if $A$ and $B$ are bounded operators with trace-class commutator $AB-BA$, then

$$\backslash det\; e^A\; e^B\; e^\{-A\}\; e^\{-B\}\; =\; \backslash exp\; \backslash operatorname\{Tr\}\; (AB-BA).$$

{{See also |Szegő limit theorems}}

Let $H=L^2(S^1)$ and let $P$ be the orthogonal projection onto the Hardy space $H^2(S^1)$.

If $f$ is a smooth function on the circle, let $m(f)$ denote the corresponding multiplication operator on $H$.

The commutator

$$Pm(f)-m(f)P$$

is trace-class.

Let $T(f)$ be the Toeplitz operator on $H^2(S^1)$ defined by

$$T(f)\; =\; Pm(f)P,$$

then the additive commutator

$$T(f)\; T(g)\; -\; T(g)\; T(f)$$

is trace-class if $f$ and $g$ are smooth.

Berger and Shaw proved that

$$\backslash operatorname\{tr\}(T(f)\; T(g)\; -\; T(g)\; T(f))\; =\; \{1\backslash over\; 2\backslash pi\; i\}\; \backslash int\_0^\{2\backslash pi\}\; f\; \backslash ,\; dg.$$

If $f$ and $g$ are smooth, then

$$T(e^\{f+g\})T(e^\{-f\})\; T(e^\{-g\})$$

is in $G$.

Harold Widom used the result of Pincus-Helton-Howe to prove that

$$\backslash det\; T(e^f)\; T(e^\{-f\})\; =\; \backslash exp\; \backslash sum\_\{\; n>0\}\; na\_n\; a\_\{-n\},$$

where

$$f(z)\; =\backslash sum\; a\_n\; z^n.$$

He used this to give a new proof of Gábor Szegő's celebrated limit formula:

$$\backslash lim\_\{N\backslash to\; \backslash infty\}\; \backslash det\; P\_N\; m(e^f)\; P\_N\; =\; \backslash exp\; \backslash sum\_\{\; n>0\}\; na\_n\; a\_\{-n\},$$

where $P\_N$ is the projection onto the subspace of $H$ spanned by $1,z,\backslash ldots,z^N$ and $a\_0=0$.

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 bosons and fermions 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 unitary matrices.

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 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).

• {{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}}

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Category:Determinants

Category:Fredholm theory

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