On Determinant Expansions for Hankel Operators

Abstract Let w be a semiclassical weight that is generic in Magnus’s sense, and (pn)n=0∞ ({p_n})_{n = 0}^\infty the corresponding sequence of orthogonal polynomials. We express the Christoffel–Darboux kernel as a sum of products of Hankel integral operators. For ψ ∈ L∞ (iℝ), let W(ψ) be the Wiener-Hopf operator with symbol ψ. We give sufficient conditions on ψ such that 1/ det W(ψ) W(ψ−1) = det(I − Γϕ1 Γϕ2) where Γϕ1 and Γϕ2 are Hankel operators that are Hilbert–Schmidt. For certain, ψ Barnes’s integral leads to an expansion of this determinant in terms of the generalised hypergeometric 2mF2m-1. These results extend those of Basor and Chen [2], who obtained 4F3 likewise. We include examples where the Wiener–Hopf factors are found explicitly.

Given a trace class Hankel operator Γ, the spectrum consists of 0 and a sequence of eigenvalues λ j , listed according to algebraic multiplicity, such that ∞ j=0 |λ j | converges.Then we define the Fredholm determinant of Γ by det(I + Γ) = ∞ j=0 (1 + λ j ).For Hilbert-Schmidt Γ, we define the Carleman determinant by det 2 (I + Γ) = ∞ j=0 ((1 + λ j )e −λj ).The purpose of the present paper is to compute Fredholm determinants such as det(I + Γ φ ), using operator theory and tools from linear systems.
Self-adjoint bounded Hankel operators have been characterized up to unitary equivalence by the results of [20].The methods of [20] emphasized the importance of linear systems, and in this paper, linear systems are used to obtain expansions of the Fredholm determinant det(I − Γ φ1 Γ φ2 ).In section 6, we consider Wiener-Hopf factorizations which lead to Barnes's integrals as in (1.5), so that φ 1 and φ 2 have explicit expansions in terms of exponential bases.When interpreted with suitable linear systems, these formulas give expansions of det(I −Γ φ1 Γ φ2 ) in terms of the generalised hypergeometric n F m .These results extend those of Basor and Chen [2], who obtained 4 F 3 likewise.In section 7, we make specific choices of ψ and interpret our results in particular examples.
Example 1.2.In the theory of random Hermitian matrices, the following example arises frequently.Let w 0 (x) be a continuous, positive and integrable weight on (0, b).Then we can take Z b > 0 such that w 0 (x j )dx j (1.9) gives a probability measure on (0, b) n .In (1.11), we identify Z b with a Hankel determinant.
For a bounded and measurable function f : R → C, we define the linear statistic n j=1 f (x j ) and consider the exponential moment generating function 1≤j<k≤n (x j − x k ) 2 n j=1 w 0 (x j )dx j . (1.10) In particular, with f (x) = − log(λ − x), we have p n (λ) = E n j=1 (λ − x j ), which is a monic polynomial of degree n.Moreover, Heine [13] showed that (p n (λ)) ∞ n=0 is the sequence of monic orthogonal polynomials with respect to the weight w 0 .We introduce h j = p j (x) 2 w(x) dx.Then the Hankel determinant x j+k w 0 (x) dx n−1 j,k=0 (1.11) satisfies (1.12) and Z b = D n [w 0 ].In section 3, we consider how Fredholm determinants are related to finite Hankel determinants det[ν(j + k)] n−1 j,k=0 when the weight w 0 is semiclassical in Magnus's sense [19].Our results continue the analysis by Tracy and Widom [32].

Linear systems and associated Hankel operators
The results of this section enable us to use linear system methods to compute Fredholm determinants of Hankel operators.For a complex separable Hilbert space H, we let L(H) = L ∞ (H) be the space of bounded linear operators on H with T the usual operator norm of T ∈ L(H), and L 1 (H) the ideal of trace class operators; then for 1 ≤ p < ∞, let L p (H) be the ideal of operators such that the Schatten Let L j (x) = (j!) −1 e x (d/dx) j (x j e −x ) be the Laguerre polynomial of order 0 and degree j; then (e −x/2 L j (x)) ∞ j=0 gives an orthonormal basis of L 2 (0, ∞).Taking the Laplace transform of the (e −x/2 L j (x)) ∞ j=0 , we obtain an orthonormal basis for the space H 2 (C + ), namely With N = {1, 2, . . .}, we introduce the standard Hilbert sequence space ℓ 2 (N ∪ {0}), with the standard orthonormal basis (e n ) and introduce the usual shift operator by the operation Se n = e n+1 on ℓ 2 (N∪{0}).
There is an unitary map We have unitary maps between the Hilbert spaces where the top arrow is the Mellin transform, the maps down on the left is the change of variables x = e −ξ for 0 < x < 1 and ξ > 0, and the bottom arrow across is the expansion in terms of the Laguerre basis.
The diagonal arrow is the Laplace transform, and the right downward arrow is given by expansion with respect to (2.1).
There are several equivalent expressions for the Hilbert-Schmidt norm of Hankel operators that appears here.Suppose that φ 1 , φ 2 ∈ L 2 (0, ∞), and extend them to L 2 (−∞, ∞) by letting φ j (u) = 0 for all u < 0. Then by a simple Fourier transform calculation as in [5] 2α Let C ∞ c be the space of infinitely differentiable functions that have compact support.
Lemma 2.1.(Basor, Tracy [6]) Suppose momentarily that f ∈ C ∞ c is real and even, so f (x) = f (−x).Then the Mellin transform f * and the Fourier cosine transform C(f ) of the function f satisfy Proof.The fractional derivative has Mellin transform where f * (s) is the usual Mellin transform of f .Hence by the Plancherel formula for the Mellin transform Proposition 2.2.The following is a commuting diagram of linear isometries, in which the top arrow is the Fourier cosine transform, and the left downwards arrow is the Mellin transform.
{f : We regard H 2 (C + ) as a closed linear subspace of L 2 (iR), and let Given c ∈ L ∞ , suppose that Γ c = P + M c J is a bounded Hankel operator.Then by Nehari's and Fefferman's theorems [23], there exists ψ ∈ L ∞ (iR) such that (2.9) Note that ψ determines c up to an additive constant; adding a constant α to c does not change ψ of Γ c .
See [23] Let H = H 2 (C + ) be the state space and let D(A) = {g(s) ∈ H : sg(s) ∈ H} with the graph norm.
Then we introduce the linear system (−A, B, C) by (2.10) The semigroup (e −tA ) t>0 operates by multiplication on the state space and is strongly continuous, so Hence by Lemma 2.2 of [23], (2.12) We introduce From the expansion of φ(t + u) as a series of rank one kernels e −ζj(u+t) , we deduce that Γ φ is trace class the simplest way to do this is by selecting f (x) = e −ξx and g(x) = e −ζx , so that (2.15) Also, we deduce that ) (2.17) (ii) Hence we can write Any bounded Hankel integral operator generates a sequence of moments, in the following sense.For φ ∈ L 2 (0, ∞), let Γ φ be the Hankel integral operator and introduce the moment sequence Magnus has characterized the moment sequences that arise as (µ n = S x n w(x) dx) for a semi classical weight on some subset of C ∪ {∞}, as we discuss in the next section.

From orthogonal polynomials to Hankel determinants
Let (p n (x)) ∞ n=0 be the sequence of monic orthogonal polynomials of degree n for some continuous and positive weight w 0 (x) on (0, b), given by the recurrence relation Then the Christoffel-Darboux formula gives so that Q n is an integrable operator.We show also that for suitable weights, Q n is a sum of products of Hankel operators.
Definition 3.1.(Magnus, [19]) (i) Let F (z) = (z − x) −1 w 0 (x) dx be the Cauchy transform of the weight w 0 on E = (0, b).The weight is said to be semi-classical if there exist polynomials U, V, W with Equivalently, the moments µ k = x k w 0 (x)dx satisfy a recurrence relation m j=0 (νξ j + η j )µ j+ν = 0 (ν = 0, 1 . . .). (3.4) for some ξ j , η k ∈ C given by the coefficients of V, W , where m is the maximum of the degrees of V and W .
(ii) A pair of polynomials (2V, W ) is said to be generic if W has degree m where m ≥ 2, the degree of V is less than m, W has m simple zeros α j and 2V /W has all residues 2V (α j )/W ′ (α j ) that are not integers.
Theorem 3.2.Let w 0 be a positive and continuous semiclassical weight on [0, ∞) that corresponds to a generic pair (2V, W ).
(iii) For f (x) = βϑ(x − t) with Re β > 0 and λ = 1 − e −β , the moment generating function of the random variable ♯{j : x j > t} subject to the probability (1.9) is given by where the scattering functions are shifted to Φ t (x) = Φ(x + t) and Ψ t (x) = Ψ(x + t).
Proof.(i) Magnus [19] shows that for each such polynomial pair, there exists a weight w 0 with Cauchy transform F and a polynomial U such that W F ′ = 2V F + U .From (11) of [19], we have W w ′ 0 = 2V w 0 .Then by (17) of [19], there exist polynomials Ω n and Θ n , and recursion coefficients a n such that with the matrices we have an ordinary differential equation where the coefficient matrix A n (x) is rational with trace zero.The three-term recurrence relation (3.1) for p n gives a positive sequence (β n ) and a real sequence (α n ) such that so we have a recurrence relation for the matrices in (3.9) where the second matrix has determinant β n > 0, hence the A n are uniquely determined.We can therefore follow the approach of section VI of [32].From the differential equation (3.9), where x−y which is rational, symmetric with respect to interchange of variables x ↔ y and symmetric with respect to matrix transpose.From the identity W w ′ 0 = 2w 0 V , and cancelling any common zeros of V and W , we deduce that W has no zeros on (0, ∞) since w 0 (x) > 0 for all x > 0 by hypothesis.Observe also that ∞ 0 x k w 0 (x)dx is finite for all k ∈ N ∪ {0}.By selecting the products of functions that depend on one variable, namely x or y, we can therefore choose φ j and φ k from among the functions in B and Y such that φ j , ψ j ∈ L 2 (0, ∞) and By integration, we obtain where q(x − y) → 0 as x → ∞ or y → ∞, so q = 0. We can select the φ j , ψ j so that , as in the Corollary, so where the final operator has a matrix kernel (iii) For Re β > 0, the point λ = 1 − e −β lies in the disc of centre 1 and radius 1 in C. Then for the step function f (x) = βϑ(x − t) we have so we have the moment generating function of the number of the x j that are greater than t.Then where each entry of the matrix is a product of Hankel operators, with scattering functions Theorem 3.2 involves a Fredholm determinant.The following result gives an equivalent expression involving finite determinants on the numerator.We introduce the block matrix Corollary 3.3.Suppose that Γ Θt ∈ L 1 and I + √ λΓ Θt is invertible.
(i) Then for any orthogonal projection P n on L 2 ((0, ∞); C 2N ) with P ⊥ n = I − P n , (ii) Let L j (x) be the Laguerre polynomial, and let P n be the orthogonal projection onto span{e −x/2 L j (x) : j = 0, . . ., n} ⊗ C 2N .
Then P n Γ Θt P n is unitarily equivalent to a finite block Hankel matrix.
Proof.(i) We have especially chosen Θ so that by Theorem 3.2(iii), we have Then the stated result follows from a determinant formula credited to Jacobi; see [2].
(ii) Hankel integral operators correspond to Hankel matrices via the Laguerre orthonormal basis of L 2 (0, ∞); see [24], page 53.(This is a special feature of the Laguerre polynomials.)To extend this to Hankel integral operators on L 2 ((0, ∞); C 2N ), we just compute the block Hankel matrix which has (2N ) × (2N ) block entries, and the cross-diagonal pattern that is characteristic of Hankel matrices.
Theorem 3.2 shows that replacing w(x) by w(x)e −βϑ(x−t) corresponds shifting Θ 0 to Θ t .The shift operation is simple to describe in terms of linear systems, as in (5.12).Unfortunately, ϑ is discontinuous, so w(x)e −βϑ(x−t) is not itself a semiclassical weight, and we cannot immediately deduce a differential equation such as 3.9 for orthogonal polynomials generated by w(x)e −βϑ(x−t) .Nevetheless, Min Chao and Chen [21] derived an ODE for gap probabilities in the Jacobi ensemble.
Hence we replace the step function by for ε > 0, and As in Theorem 3.2, we suppose that w 0 satisfies W w ′ 0 = 2V w 0 , where V, W are polynomials, and let v 0 = − log w 0 .Then there exists ε 0 > 0 such that is also generic for all real β and 0 < Im z + < ε 0 and 0 < − Im z − < ε 0 .In particular, we can replace our previous weight w 0 (x) by then we build the system of monic orthogonal polynomials (p j (x)) ∞ j=0 for the complex bilinear form is generic for all real β and 0 < ε < ε 0 ; (ii) there exists a consistent system of ordinary differential equations as in (3.8) where A(x, t; β, ε) is a proper rational function of x with trace zero, and simple poles at the zeros of W and t ∓ iε; (iii) the consistency condition holds Proof.(i) This is a direct check of the definitions.Then the modified potential v = − log w has v ′ rational, and we obtain a family of pairs of polynomials, depending upon parameters (t, ε, β).For given n, we can choose ε 0 > 0 such that the Gram-Schmidt process for the bilinear form f, g produces orthogonal polynomials of degree up to n, for all 0 < ε < ε 0 .
(ii) Magnus [19] obtains Θ n and Ω n by recursion, and one checks that the degree of Θ n is less than or equal to m, while the degree of the denominator is m + 2. From his recursion formula (20), the degree of Ω 2 n is less than or equal to 2(m + 1), so A(x, t; β, ε) is strictly proper.By Proposition 3.4(ii), we can write where the 2 × 2 residue matrices A j , A ± depend upon (β, z ± ), but not upon x.The set of singular points in the Riemann sphere C ∪ {∞} is {α 1 , . . ., α m , z ± , ∞}.
We can take z ± = t ± iε, a complex conjugate pair.Then we fix β ∈ R and some 0 < ε < ε 0 and regard t as the main deformation parameter.Then the weight is positive and continuous on E, so p j is a real polynomial and h j > 0. Since the differential equation (3.27) has only regular singular points, the monodromy is fully described in [25] by results of Schlesinger page 148 and Dekkers page 180 in terms of connections of dimension two on the punctured Riemann sphere.Schlesinger found the condition for the system to undergo an infinitesimal change in the poles {α 1 , . . ., α m ; z ± } that does not change the monodromy.Let Y be the fundamental solution matrix of (3.27), and introduce to obtain the required variation in z ∓ .
(iii) This formula follows from the equality of mixed partial derivatives where Y is the fundamental solution matrix of (3.27) and ∂/∂t = ∂/∂z + +∂/∂z − .To ensure that the differential equations are indeed consistent, we require where by Schlesinger's equations Proof.By translating z to z+t, we replace the singular points (t−iε, t+iε, α 1 , ∞) by (−iε, +iε, α 1 −t, ∞), so we have variation in only one pole.Then we can apply known results from [15] and [17] to reduce the compatibility condition (3.29) to a Painlevé VI ordinary differential equation.
Remark 3.6.(i) Chen and Its [7] showed that the Hankel determinant D[w] gives the isomonodromic τ function for the system of Schlesinger equations that describe the isomonodromic deformation of (3.27) with respect to the position of the poles.The Schlesinger equations may be solved in terms of the Θ-function on a hyperelliptic Riemann surface, as in [18].The solutions to the monodromy preserving differential equations have singularities which are poles, except for the fixed singularities.Previously, Magnus [19] had found conditions for the system (3.8) to undergo an isomonodromic deformation, and obtained examples that realise the nonlinear Painlevé VI equation as (3.29).
(iii) By taking ε → 0+, have z ± → t and . (3.39) In section 7, we consider the behaviour of this determinant for large n.
Proof.(i) We take the norm to be Evidently C 2 is a subspace of the Banach algebra H ∞ of bounded functions on the strip {z : | Im z| < ε}, hence C 2 is an integral domain.
(ii) The Hankel integral operator with kernel φ(s + t) on L 2 (0, ∞) has Hilbert-Schmidt norm satisfing where we have used Plancherel's formula.By Cauchy's integral formula for derivatives, we have Hence Γ( f ) is a Hilbert-Schmidt operator.
Let ψ be typical element of C 2 such that ψ(z) → 1 as z → ±∞ along the imaginary axis and such that ψ has no zeros on the imaginary axis.The function Then ψ has the form where (1) w j are the zeros of ψ(z) for | Re z| ≤ ε/2, (2) k is the winding number of the contour {ψ(iξ) : −∞ ≤ ξ ≤ ∞}, (3) χ + is holomorphic and bounded on Re z ≥ −ε/2 and (4) χ − is holomorphic and bounded on Re z ≤ ε/2 with See also the results of Rappaport from [27].
The spaces H ∞ ({s : Re s < ε}) and H ∞ ({s : −ε < Re s}) have intersection C by Liouville's theorem, so χ + and χ − are unique up this additive constant.If ψ ∈ G(C 2 ), then ψ has no zeros and the middle factor is absent, but we are left with the initial factor incorporating the winding number.
Let Q be an orthogonal projection on L 2 ((0, ∞); dx), and introduce the complementary spaces For g ∈ L ∞ , let M g ∈ L(L 2 ) be the multiplication operator M g : h → gh.Then we introduce Lemma 4.2.Let C p be the space of g ∈ L ∞ such that Γ g ∈ L p and Γg ∈ L p , and let Proof.For g ∈ L ∞ we have M g ∈ L ∞ , and g L ∞ ≤ M g L ∞ ≤ g Cp , so the pointwise multiplication is unambiguously defined.Conversely, suppose that g, h ∈ C p , and observe that leading to identities such as The ideal property of the Schatten norm gives and similar inequalities for each entry of (), hence the norm satisfies the submultiplicative property.
Let A 2 be the subalgebra of C 2 consisting of f ∈ C 2 such that f is bounded and holomorphic on the right half plane, and let A * 2 be the subalgebra of C 2 consisting of f ∈ C 2 such that f (z) = ḡ(−z) for some g ∈ A 2 .Note that A * 2 ∩ A 2 = C1 by Liouville's theorem.The following result describes ψ ∈ G(A * 2 )G(A 2 ) that has no imaginary zeros, but may have zeros elsewhere.For G a group, we write {X, Y } = XY X −1 Y −1 for the multiplicative commutator.

Wiener-Hopf determinant
This section contains the main theoretical result, as follows.(

5.2)
There are three particular cases that arise under the following hypotheses: Any pair of these conditions implies the other one.
Proof.By the Lemma 4.3, we can choose ε ′ > 0 such that (5.5) Now the operators W (ψ ± ) are invertible, and W (ρψ so taking the determinant of the inverse of the right-hand side (5.7) We also have Taking the unitary conjugation by the Fourier transform, we have Γ(f ) → Γ φ1 and Γ(g) → Γ φ2 , where − 1 e −iξx dξ; (5.11) the difference in signs ±ξ in the quotients reflecting the tilde on Γ(f ).
Suppose that the linear system (−A, B j , C) realises φ j .Then the matrix system is determined by the spectrum of the scalar-valued Hankel operator Γ φ1 .The nature of the spectrum is determined in [20,24].
Finally, one considers the cases (i), (ii) and (iii) as they apply to 0 U V 0 .
As in Corollary 3.3, we can reduce the Fredholm determinant of Hankel operators to related determinants.
Let P and Q be orthogonal projections on L 2 (0, ∞) such that P + Q = I.Then Self-adjoint block Hankel matrices have been characterized up to unitary equivalence, as in Theorem 2 of [20].
Corollary 5.2.Let a j , b j , c j , d j ∈ (0, ∞) and where the zeros and poles satisfy m j=1

.21)
Then there exists a linear system as in (5.12) such that (i) Also, φ 1 and φ 2 are real.
Let s = iξ and z = e −x where Re x > 0 so |z| < 1.We consider where we have used the formula Γ(w)Γ(1 − w) = πcosec πw; now we take an integral round a semicircular contour in the left half plane and sum over the residues at poles near the negative real axis of s to obtain where we have picked out the factor cosec π(s j + s)/Γ(1 − a j − s) that contributes the pole, so and 1 stands for the omitted term in the denominator, and we have written this expression in terms of the generalized hypergeometric functions, as in [11], page 182.There is a similar formula for φ 2 (z) in which (c j , d j , b j , a j ) replaces (a j , b j , d j , c j ) Without loss of generality, we suppose a 1 < a 2 < • • • < a m , so taking the term from Res(−a 1 ), we have and likewise with c 1 < c 2 < • • • < c m , taking the term from Res(−c 1 ), we have We replace the doubly indexed family of powers by a singly indexed sequence by introducing j = mk + r and λ j = a r+1 + k and η j = c r+1 + k, thus obtaining the sequences (λ j ) ∞ j=0 = (a 1 , a 2 , . . ., a m , a 1 + 1, a 2 + 1, . . ., a m + 1, a 1 + 2, . . .), ( where there is a recurring pattern of length m.With the coefficients given above, suitably re-indexed, let γ j e −ηj x .(6.9) Proposition 6.1.Suppose that φ 1 and φ 2 are as in (6.7), (6.8) and (6.9).Then the determinant from Corollary 5.2 is given by Proof.We have a series of rank-one kernels where |ξ j |e −λj x /|λ j | converges, so Γ φ1 is trace class on L 2 (x, ∞).Then we introduce the linear systems and likewise (−A 2 , B 2 , C 2 ) when we combine them into (−A, B, C) with scattering function and write Φ(t + 2x) = Φ (x) (t).We also consider the operator To help compute the Fredholm determinant of R x , we also let Ξ x : L 2 (0, ∞) → ℓ 2 and Θ x : L 2 (0, ∞) → ℓ 2 be defined by We observe that Θ x is trace class, and likewise Ξ x is trace class since j ξ j e −λj x converges absolutely.Whereas (e −λj t ) ∞ j=0 is not an orthogonal basis, the map Θ x is injective by Lerch's theorem and span{e −λjt , j = 0, 1, . . .} is dense in L 2 (0, ∞).

Then we observe that Γ Φ
hence With respect to the standard orthonormal basis of ℓ 2 , we have a matrix representation for the top right corner of R x as in (6.16), and for the bottom left corner of R x as in (6.16).
Whereas R 2,1 is not quite the transpose of R 1,2 , the matrices have a high degree of symmetry which becomes clear when we make our expansion of the determinant.For a finite subset S of N ∪ {0}, let ♯S be the cardinality of S, and ∆ j∈S (λ j ) be Vandermonde's determinant formed from λ j with j ∈ S naturally ordered.For an infinite matrix V , and T ⊂ N ∪ {0}, let det[V ] S×T be the determinant formed from the submatrix of V with rows indexed by j ∈ S columns indexed by ℓ ∈ T , naturally ordered.Then by the Cauchy-Binet formula.Then by Cauchy's formula, the summand involving ℓ∈S,j∈T (λ ℓ + η j ) (6.24) Hence we have the determinant expansion of .
The set of all divisors on Ω forms an additive group D(Ω).For each meromorphic function, we associate the divisor given by the sum of nδ z for each zero of order n at z, and −mδ p for each pole of order m at p.For Γ-functions, it is convenient to have the following shorthand.For s ∈ C we write C ∞ c (R; R), and consider the Schrödinger equation with even potential q.There exist and even solution f + and and odd solution f − such that so θ ± is the phase shift.Let the reflection coefficient be R(ξ) = (−1/2)(e −iθ+(ξ) + e −iθ−(ξ) ).Then with −∞ e iξx R(ξ) dξ, we introduce Γ φ and as in [10].In section 5.7 of [10], the authors interpret ϑ as a theta function on an infinite-dimensional torus, and obtain series expansions for the determinant.In the current paper, we use the exponential series (2.13) and (6.11) instead, which lead to formulas (6.24) which resemble those on 5.7 and 5.8 in [10].
In particular, consider the Schrödinger equation Then the scattering amplitude is the coefficient of f (x) for large x when e −iλx is the scattering function. Then which has divisor, in terms of s = iξ, Example 7.2.Meier's G-function [11] page 206 is x s ds (7.5) where we take all the a j , b j in {s : 0 < Re s < 1} with degree p + q − 2m − 2n, which we take to be negative.Then the divisor for the quotient of Gamma functions in the integrand is which is holomorphic on −ν < Re s < 1 − ν; see [28].Also, for s = η + iξ and −1 < η < −1/2, we have where here γ is Euler's constant.Hence the Cauchy transform diverges at some points with |z| = 1.Also ∞ j=0 jµ 2 j diverges, so the Hankel moment matrix is not Hilbert-Schmidt.
Nevertheless, the Hankel moment matrix [µ j+m ] ∞ j,m=0 defines a bounded linear operator on ℓ 2 .To see this, we transform to Hankel integral operators on L 2 (0, ∞) via the Laguerre functions.With J 0 standing for Bessel's function of the first kind of order zero, the orthonormal Laguerre functions in L 2 (0, ∞) satisfy We introduce the scattering function which we can express as an integral so we have, on substituting s We now express φ ν as the scattering function of a continuous time linear system.Let the state space be H = L 2 (0, ∞), with dense linear subspace D(A) = {f ∈ H : tf (t) ∈ H}.Then for −1/2 ≤ Re ν < 1/2, we introduce the linear system (−A, B ν , C) by the corresponding scattering function is Then we introduce the operator which is the integral operator on L 2 (0, ∞) that has kernel Evidently R ν is the composition of Hilbert's Hankel operator with kernel 1/(τ + t) and multiplication by (1/2 + τ ) ν−1 , so R ν is bounded on L 2 (0, ∞).Operators of this form were considered by Howland [16].

Application of equilibrium problem to linear statistics
In this section we consider with particular emphasis on w 0 = e −nv0 where v 0 ∈ C 2 is convex and f (x) = ϑ(x − t) is a step function.
Then the exponent in numerator of the expression (1.10) involves We regard this as the electrostatic energy associated with n positive and equal charges on a line, subject to an electrical field.The following result [26] extends a familiar result to the case of discontinuous fields.U n is the Chebyshev polynomial of the second kind of degree n such that U n (cos φ) = sin(n + 1)φ/ sin φ.
Then by the Lemma 8.2, we have where the branch of the square root is chosen so that the integrals converge to zero as z → ∞, we deduce that ρ from (8.4) satisfies where C is chosen so that

1 .
Introduction Definition 1.1.(i) Let φ ∈ L 2 (0, ∞).Then the Hankel operator with scattering function φ is the integral operator Γ φ f (x) = ∞ 0 φ(x + y) f (y) dy (1.1) be the right half-plane and let H 2 (C + ) be the Hardy space of holomorphic functions f on C + such that sup x>0 ∞ −∞ |f (x + iξ)| 2 dξ is finite.By the Paley-Wiener Theorem, the Mellin transform gives a unitary transformation that restricts to the orthogonal subspaces

8 )
We show that trace class Hankel operators on Hardy space H 2 (C + ) have a matrix representation with respect to reproducing kernels on the state space.Let C + = {z ∈ C : Re z > 0} and C − = {z ∈ C : Re z < 0}; then we introduce the usual Hardy spaces H 2 (C + ) and H 2 (C − ) which are related by the unitary involution J :

Proposition 4 . 1 .
(i) Then C 2 is a commutative and unital Banach algebra under the usual pointwise multiplication,