The spectral density of Hankel operators with piecewise continuous symbols
Emilio Fedele


Spectral density of Hankel operators]The spectral density of Hankel operators with piecewise continuous symbols

Department of Mathematics, King's College London, WC2R 2LS, London.
mailto:emilio.fedele@kcl.ac.ukemilio.fedele@kcl.ac.uk
[
    Emilio Fedele
    December 30, 2023
=====================


In 1966, H. Widom proved an asymptotic formula for the distribution of eigenvalues of the N× N truncated Hilbert matrix for large values of N. In this paper, we extend this formula to Hankel matrices with symbols in the class of piece-wise continuous functions on the unit circle. Furthermore, we show that the distribution of the eigenvalues is independent of the choice of truncation (e.g. square or triangular truncation).


Introduction
General setting and first results.
Given an (essentially) bounded function ω, called a symbol, on  the unit circle ={v ∈ | v=1}, the associated Hankel matrix, (ω), is the (bounded) operator on , _+={0,1,2,…}, whose “matrix” entries are 
    (ω)_j,k=ω(j+k),   j, k ≥ 0,
 where ω denotes the sequence of Fourier coefficients 
    ω(j)=∫_0^2πω(e^ i θ)e^- i jθdθ/2π,    j ∈.
 The matrix (ω) is always symmetric. In particular, it is self-adjoint if and only if ω is real-valued. For instance, this is the case when ω satisfies the following symmetry condition

    ω(v)=ω(v),   v ∈.

In this paper, we consider symbols in the class of the piece-wise continuous functions on , denoted by PC(), i.e. those symbols ω for which the limits 

    ω(z+)=lim_→ 0+ω(z e^i ),   ω(z-)=lim_→ 0+ω(z e^-i ),

 exist and are finite for all z ∈. The points z ∈ for which the quantity 
    κ_z(ω)=ω(z+)-ω(z-)/2≠ 0
 are called the jump discontinuities  of ω and κ_z(ω) is the half-height of the jump of the symbol at z. Due to the presence of jump discontinuities, Hankel matrices with these symbols are non-compact. The compactness of  and the existence of the limits in (<ref>) can be used to show that the sets 
    Ω_s={z ∈|κ_z(ω)>s},    s>0,
 are finite and so the set of jump-discontinuities of ω, denoted by Ω, is at most countable. Furthermore, if the symbol satisfies (<ref>), Ω is symmetric with respect to the real axis and for any z ∈Ω 
    κ_z(ω)=-κ_z(ω),
 whereby we obtain that  κ_z(ω)=κ_z(ω), and at z=± 1, κ_z(ω) is purely imaginary.

Hankel matrices with piece-wise continuous symbols still attract attention in both the operator theory and spectral theory community, see for instance <cit.> and references therein. S. Power, <cit.>, showed that the essential spectrum of such matrices consists of bands depending only on the heights of the jumps of the symbol and gave the following identity:

    (ω)=[0, -iκ_1(ω)]   ∪[0, -iκ_-1(ω)]∪
       ⋃_z ∈Ω\{± 1}[-i(κ_z(ω)κ_z(ω))^1/2,i(κ_z(ω)κ_z(ω))^1/2],
 
where the notation [a,b], a,b ∈ denotes the line segment joining a and b. Assuming that the symbol has finitely many jumps and, say, it is Lipschitz continuous on the left and on the right of the jumps, in <cit.>, a more detailed picture is obtained for the absolutely continuous (a.c.) spectrum of (ω)=√((ω)^∗(ω)), where the following formula is obtained 
    (ω)=⋃_z ∈Ω[0, κ_z(ω)].
 Furthermore, it is shown that each band contributes 1 to the multiplicity of the a.c. spectrum.

 
First examples of symbols fitting in this scheme are the following 

    γ(e^iθ)=iπ^-1 e^-iθ(π-θ),  ψ(e^iθ)=2 _E(e^iθ),  θ∈ [0, 2π).

where _E is the characteristic function of the set E={θ∈ [0,  2π): cosθ>0}. It is clear that both γ, ψ∈ PC(), and their jumps occur at z=1 and z=± i respectively and κ_1(γ)=i, κ_± i(ψ)=∓ 1. Simple integration by parts shows that 
    γ(j)=1/π(j+1),  ψ(j)=2sin(π j/2)/π j,    j≥ 0,
 with the understanding that ψ(0)=1. Power's result in (<ref>) in these cases gives 
    (γ)=[0,  1],   (ψ  )=[-1,  1].
 The matrix (γ), known as the Hilbert matrix, has simple a.c. spectrum coinciding with the interval [0,1] and a full spectral decomposition was exhibited in <cit.>.  In <cit.>, the authors perform a more detailed spectral analysis of (ψ) and show that its spectrum is purely a.c. of multiplicity one and coincides with the interval [-1,  1].


For N≥ 1, let ^(N)(ω) be the N× N Hankel matrix 
    ^(N)(ω)={ω(j+k)}_j,k= 0^N-1.
 We wish to give a description of the relationship between the spectrum of the infinite matrix (ω) and that of its truncation ^(N)(ω). More specifically:

    
  * for a non-self-adjoint Hankel matrix, we study the distribution of the singular values of ^(N)(ω) inside the spectrum of (ω);
    
  * in the self-adjoint setting, we study the distribution of the eigenvalues of ^(N)(ω) inside the spectrum of (ω).

To do so, for a non-self-adjoint Hankel matrix (ω) we study the asymptotic behaviour of the singular-value counting function

    (t;^(N)(ω))=#{n: s_n(^(N)(ω))>t},   t>0,

as N →∞. Here {s_n(^(N)(ω))}_n≥ 1 is the sequence of singular values of ^(N)(ω). In particular, we study the logarithmic spectral density of (ω), defined as

    _□(t;(ω)):=lim_N →∞(t; ^(N)(ω))/log(N).

 
For a self-adjoint (ω), its spectrum, (ω), is a subset of the real line and so we look at how the positive and negative eigenvalues of ^(N)(ω) distribute inside (ω). To this end, we analyze the behaviour of the eigenvalue counting functions 
    _±(t;^(N)(ω))=#{n: λ^±_n(^(N)(ω))>t},   t>0,
 as N →∞. Here {λ^±_n(^(N)(ω))}_ n≥ 1 are the sequences of positive eigenvalues of ±^(N)(ω) respectively. In this setting, we study the functions

    _□^±(t;(ω)):=lim_N →∞_±(t; ^(N)(ω))/log(N).

Similarly to the non-self-adjoint setting, we call the function ^+_□ (resp. ^-_□) in (<ref>) the positive (resp. negative) logarithmic spectral density of (ω). 

The □ appearing as an index in the definitions of the logarithimic spectral densities in (<ref>) and (<ref>) has been chosen to stress the fact that, a priori, these quantities depend on our choice to truncate the infinite matrix (ω) to its upper N× N square. Furthermore, the terminology we use for the functions _□ and ^±_□ comes from the fact that we are only studying a logarithmically-small portion of the singular values (or eigenvalues) of the matrix ^(N)(ω). Their definitions are motivated by the results obtained by Widom (see <cit.>) for the Hilbert matrix (γ), where he showed that

    _□(t;(γ))=𝖼(t),
    _□^-(t;(γ))=0,  ^+_□(t;(γ))=𝖼(t).

Here 𝖼(t):=0 whenever t ∉ (0,1) and 

    𝖼(t):=1/π^2(t)=1/π^2log(1+√(1-t^2)/t),    t ∈ (0,1].

We note that a factor of 2π is missing in the statement of <cit.>. The aim of this paper is to extend (<ref>) to a general symbol ω∈ PC(). In particular, for a non-self-adjoint Hankel matrix, we aim to show that

    _□(t;(ω))=∑_z∈Ω𝖼( tκ_z(ω)^-1),

where 𝖼 is the function defined in (<ref>). Recall that the symbol ψ defined in (<ref>) has jumps at ± 1 whose half-height is κ_± i(ψ)=∓ 1, so for the Hankel matrix (ψ) the formula (<ref>) yields

    _□(t;(ψ))=2 𝖼(t).

 For self-adjoint Hankel matrices, we extend the result in (<ref>) to symbols ω∈ PC() satisfying (<ref>) and obtain

    ^±_□(t;(ω))   =𝖼(tκ_1(ω)^-1)_±(-iκ_1(ω))+𝖼(tκ_-1(ω)^-1)_±(-iκ_-1(ω))   +∑_z∈Ω^+𝖼(tκ_z(ω)^-1),

where Ω^+={z∈Ω | z>0}, and _± is the indicator function of  the half-line (0, ±∞). Again, the function 𝖼 has been defined in (<ref>).  In particular, for the symbol ψ in (<ref>), we obtain that 
    ^±_□(t;(ψ))=𝖼(t).


A natural question that we also address here is that of the universality of  the limits in (<ref>) and (<ref>). In other words, we investigate whether they depend on the choice of “regularisation” of the matrix (ω). For instance, the main results of this paper, Theorems <ref> and <ref> below, tell us that the singular values of the matrix ^(N)(ω) and of the regularised matrix

    _N(ω)={e^-j+k/Nω(j+k)}_j,k≥ 0,   N≥ 1,

have the same distribution for large values of N. 


Schur-Hadamard multipliers
The truncation of a matrix to its finite N× N upper block and the “matrix regularisation" in (<ref>) are examples of Schur-Hadamard multipliers, defined below.

For a bounded sequence (τ(j,k))_j,k≥ 0, called a multiplier, and a bounded operator A on , the Schur-Hadamard multiplication of τ and A is the operator on , τ⋆ A, formally defined through the quadratic form

    ((τ⋆ A)e_j, e_k)=τ(j,k)(Ae_j,e_k),   j, k≥ 0,

where e_j is the j-th vector of the standard basis of . Various authors in the literature, <cit.>, have addressed the issue of establishing how properties of τ translate into the boundedness of this operation on the space of bounded operators and the Schatten classes  (for a definition see section 2 below). To do so, they have studied the operator norms

    τ_   =sup_A=1τ⋆ A,
    τ__p   =sup_A_=1τ⋆ A_,   1≤ p≤∞.

Using the duality of -classes, it is possible to show that the following identities hold 

    τ__1   =τ__∞=τ_,
    τ__p   =τ__q,   1< p,  q<∞,  p^-1+q^-1=1,

and so it is sufficient to study the boundedness of Schur-Hadamard multiplication on  for p≥ 2. The case of p=2 is somewhat trivial. In fact, the structure of _2 gives that any bounded sequence τ is a bounded Schur-Hadamard multiplier and furthermore 
    τ__2=sup_j,k≥ 0τ(j,k).
 For a general 1<p<∞,  p≠ 2 not much is known with regards to the finiteness of τ__p. However, a necessary and sufficient condition for the boundedness of a Schur-Hadamard multiplier on the space of bounded operators (and, as a consequence of (<ref>), on _1 and ) is known and can be found in <cit.>. 

For the purposes of this paper, we will consider the Schur-Hadamard multiplier τ in (<ref>) as the restriction to _+^2 of a bounded function defined on [0, ∞)^2. For N≥ 1 set τ_N(j,k)=τ(jN^-1, kN^-1). If τ is such that the sequence of τ_N satisfies the following

    sup_N≥ 1τ_N_<∞,

we say that τ induces a uniformly bounded multiplier. An easy example of such a multiplier is the N× N truncation of an infinite matrix. To see this take the function 

    τ_□(x,y)=_□(x,y),

where _□ is the characteristic function of the half-open unit square [0,1)^2. For any bounded operator A, τ_N⋆ A is the truncation to its upper N× N block and so we have that for any N ≥ 1 
    τ_N_= 1.
 We discuss some more examples of Schur-Hadamard multipliers below. 


Statement of the main results. 
As we anticipated, our main results are not only concerned to the existence of the limits in (<ref>) and (<ref>), but also with their universality. In other words, for a Hankel matrix (ω) and  a given multiplier τ, we show that under some mild assumptions on τ, see <ref>-<ref> below, the function

    _τ(t;(ω)):=lim_N→∞(t;τ_N⋆(ω))/log N,   t>0,

is independent of the choice of τ. Similarly, for a self-adjoint Hankel matrix and a multiplier τ such that τ(x,y)=τ(y,x), we show that the same is true for the functions

    _τ^±(t;(ω)):=lim_N→∞_±(t;τ_N⋆(ω))/log N,   t>0.

Note that when τ=τ_□ as in (<ref>), the functions _τ(t;(ω)) and _τ^±(t;(ω)) are precisely those defined in (<ref>) and (<ref>).

Let us state the following assumptions on τ:


  *  τ induces a uniformly bounded Schur-Hadamard multiplier, i.e. (<ref>) holds;

  *  τ(0,0)=1 and for some >0 and some β >1/2, there exists C_β>0, so that  
    τ(x, y)-1≤ C_βlog(x+y)^-β,   ∀  0≤ x, y≤;


  *  for some α>1/2 one can find C_α so that 
    τ(x,y)≤ C_αlog(x+y+2)^-α,   ∀  x,  y≥ 0.
 

Then (<ref>) is a particular case of the following:

Let τ be a multiplier satisfying <ref>-<ref>. Let ω∈ PC() and Ω be the set of its discontinuities. Then

    _τ(t;(ω))=∑_z∈Ω𝖼( tκ_z(ω)^-1)

where 𝖼(t) is the function defined in (<ref>).

Analogously for the self-adjoint case, (<ref>) is a particular case of the Theorem below:

Let τ satisfy conditions <ref>-<ref> and such that τ(x, y)=τ(y,x). Suppose ω∈ PC() satisfies (<ref>) and let Ω^+={z ∈Ω |  z>0}. Then 

    _τ^±(t;(ω))   =∑_z∈Ω^+𝖼(tκ_z(ω)^-1)   +𝖼(tκ_1(ω)^-1)_±(-iκ_1(ω))   +𝖼(tκ_-1(ω)^-1)_±(-iκ_-1(ω)),

where 𝖼(t) is the function defined in (<ref>) and _± is the characteristic function of the half-line (0, ±∞).



Remarks

    
  * It is clear that Theorems <ref> and <ref> generalise the result of Widom in <cit.> mentioned earlier in (<ref>) to any multiplier τ and, in both instances, we only describe the behaviour of a logarithmically small portion of the spectrum of τ_N⋆(ω) as most of the points lie in a vicinity of 0.
    
  * Both Theorems <ref> and <ref> deal with a rather general class of symbols and for this reason we cannot say more about the error term in the asymptotic expansion of the functions , _±. In fact, we can only write 
    (t; τ_N⋆(ω))=log(N)∑_z ∈Ω𝖼(tκ_z(ω)^-1)+o(log(N)),   N→∞.
 If, however, we were to restrict our attention to those symbols with finitely many jumps and some degree of smoothness away from them (say Lipschitz continuity), we would obtain a more precise estimate,  see <cit.>, however the trade-off would be that of making our results less general.
    
  * Studying the spectral density of operators is common to many areas of spectral analysis. In particular, our results can be put in parallel to well-known results in the spectral theory of Schrödinger operators, where the existence and universality of the density of states is a well-studied problem for a wide class of potentials, see <cit.> and <cit.> for an introduction and references therein for more on this subject. 
    
  * Both Theorems <ref> and <ref> assume that the multiplier τ induces a uniformly bounded multiplier on the space of bounded operators. However, this condition can be substantially weakened in two different ways.
    
    Firstly, we can weaken assumption <ref> on the multiplier τ by assuming that  for some finite p>1, τ induces a uniformly bounded Schur-Hadamard multiplier on , or in other words that
    
    sup_N≥ 1τ_N__p=sup_N≥ 1(sup_A_=1τ_N⋆ A_)<∞.

    However, as a trade-off, we need to impose more stringent conditions on the symbol, as the following statement shows:
    
    Suppose τ satisfies (<ref>) as well as Assumptions <ref> and <ref>. If the symbol ω can be written as 
    
    ω(v)=-i∑_z ∈Ωκ_z(ω)γ(zv)+η(v),   v∈,

    where Ω is a finite subset of , γ is the symbol in (<ref>) and η is a symbol for which (η) ∈, then (<ref>) holds. Furthermore, if τ(x,y)=τ(y,x) and ω also satisfies (<ref>), then (<ref>) holds.
    
    Secondly, we can assume that τ only induces a uniformly bounded Schur-Hadamard multiplier on the space bounded Hankel matrices, i.e. that
    
    sup_N≥ 1(sup_(ω)=1τ_N⋆(ω))<∞.

     In this case, Theorems <ref> and <ref> still hold in their generality and we have the following
    
     Let ω∈ PC() and let τ satisfy (<ref>) as well as Assumptions <ref> and <ref>. Then (<ref>) holds. Furthermore, if ω satisfies the symmetry condition (<ref>) and τ(x,y)=τ(y,x), then (<ref>) holds.
    
    We chose to make use of Assumption <ref> instead of (<ref>) and (<ref>), because there are no known necessary and sufficient conditions for a multiplier to satisfy either of them. We give specific examples of multipliers that satisfy these conditions below. 
    



Some Examples of Schur-Hadamard multipliers
[Factorisable multipliers]
If the function τ can be factorised as 
    τ(x,y)=f(x)g(y),   x,y≥ 0,
 for some bounded function f, g, then it is easy to see that it induces a uniformly bounded Schur-Hadamard multiplier in the sense of (<ref>), and furthermore 
    sup_N≥ 1τ_N_≤f_∞g_∞.
 As it was pointed out earlier in (<ref>), the truncation to the upper N× N square is an example of such a multiplier. Another example is given by choosing the function τ_1(x, y)=e^-(x+y)=e^-xe^-y. This induces the regularisation in (<ref>) and it is immediate to see that 
    sup_N≥ 1(τ_1)_N_=1.
 Furthermore, τ_1 satisfies the assumptions <ref> and <ref> and so Theorems <ref> and <ref> hold.

[Non-examples] In stark contrast to the square truncation in (<ref>), the so-called “main triangle projection" induced by the function 

    τ_2(x,y)=_[0,1)(x+y)

is not uniformly bounded on the bounded operators, see <cit.>, where it was shown that 
    sup_A=1(τ_2)_N⋆ A= π^-1log(N)+o(log(N)),   N →∞.
 However, τ_2 is uniformly bounded on any Schatten class ,  1<p<∞, see <cit.>, and so Proposition <ref> holds.

Proposition <ref> shows that Theorems <ref> and <ref> still hold in the case that the Schur-Hadamard multiplier is only uniformly bounded on the set of bounded Hankel matrices. An example of such a multiplier is given by the indicator function, τ_β, γ, of the region 
    Ξ_β, γ={(x, y)∈ [0,1]^2 | x≤ -β y+ γ},   β, γ∈.
 Even though τ_β, γ does not induce, in general, a uniformly bounded Schur-Hadamard multiplier, it has been shown in <cit.> that this is the case on the set of bounded Hankel matrices for β≠ 1, 0 and any γ (at β=1 and γ=1, τ_1, 1 reduces to the multiplier τ_2 considered above). With this at hand, an appropriate choice of the parameters β and γ gives (<ref>) and (<ref>).

[General Criterion]
For more complicated functions, the following criterion can be of help. Let Σ⊂ and m be a measure on Σ. Suppose that for the function τ we can write 
    τ_N(j,k)=τ(j/N, k/N)=∫_Σe^-it(j+k)f_N(t)dm(t),  ∀ j,  k≥ 0
 for some functions f_N∈ L^1(Σ, m) so that sup_N≥ 1f_N_L^1(Σ, m)<∞. It is not hard to check that τ induces a uniformly bounded Schur-Hadamard multiplier and that 
    sup_N≥ 1τ_N_≤sup_N≥ 1f_N_L^1(Σ, m).
 Using this it is possible to show that the function

    τ_3(x,y)=(1-(x+y))_[0,1)(x+y),

induces a uniformly bounded Schur-Hadamard multiplier, since one has the following representation

    τ_3(j N^-1,k N^-1)=∫_0^2πe^-i t (j+k)F_N(e^it)dt/2π,   j,  k≥ 0,

where F_N in (<ref>) denotes the N-th Fejér kernel.

The multipliers induced by the functions 

    τ_1(x,y)=e^-(x+y),
    τ_2(x,y)=_[0,1)(x+y),
    τ_3(x,y)=(1-(x+y))_[0,1)(x+y)

are related to the Abel-Poisson, Dirichlet and Cesaro summation methods respectively and share some of the properties of the operators of convolution with the respective kernels, see Section 3 for more on the Poisson kernel.



Outline of the proofs
To prove Theorems <ref> and <ref> we use a similar approach to the one in <cit.> and <cit.> and combine abstract results concerned with the general properties of the functions , _± (see Section 2) and more hands-on function theoretic ones that are specific to the theory of Hankel matrices, see Section 3. 

To prove Theorem <ref>, we firstly assume that the set of jump-discontinuities, Ω, of the symbol ω is finite and we write

    ω(v)=-i∑_z ∈Ωκ_z(ω)γ(zv)+η(v),   v∈,

where γ is the symbol in (<ref>) and η is a continuous function on . The analysis of _τ(t;(ω)) then proceeds with the study of each summand appearing in (<ref>) and the interactions this has with all the others. In particular, Assumption <ref> allows us to disregard the contribution coming from the matrix (η), i.e. it gives that 
    _τ(t;(ω))=_τ(t;∑_z∈Ωκ_z(ω)(γ_z)),
 where γ_z(v)=-iγ(zv),  v ∈.
 The invariance of the functions _τ with respect to the choice of multiplier, proved in Theorem <ref>, gives that 
    _τ(t;∑_z∈Ωκ_z(ω)(γ_z))=_τ_1(t;∑_z∈Ωκ_z(ω)(γ_z))
 where the multiplier σ_1(x,y)=e^-(x+y) is given in the Example <ref> above, and it is shown to induce the regularisation in (<ref>), i.e. (τ_1)_N⋆(ω)=_N(ω). For the multiplier τ_1, we explicitly show that the operators (γ_z) are mutually “almost orthogonal" in the sense that if z≠ w ∈Ω, then both  
    _N(γ_z)^∗_N(γ_w),  _N(γ_z)_N(γ_w)^∗
 are trace-class. From here, Theorem <ref>, gives that each jump contributes independently, or in other words that we can write 
    _τ_1(t;∑_z∈Ωκ_z(ω)(γ_z))=∑_z ∈Ω_τ_1(t;κ_z(ω)(γ_z)).
 We note here that the above is another instance of the general fact that jumps occurring at different points of the unit circle contribute independently to the spectral properties of the operator (ω). For this reason, we follow the terminology used by the authors of <cit.> and we refer to this fact as the “Localisation Principle".
 
 Finally, using once again the Invariance Principle, Theorem <ref>, and the result of Widom in (<ref>), we obtain the identity  (<ref>) for a symbol ω with finitely many jumps.

The proof of Theorem <ref> roughly follows the same outline. However, instead of writing the symbol ω as in (<ref>), we make use of the symmetry of the set of jump-discontinuities, Ω, to decompose it as follows

    ω(v)=κ_1γ_1(v)+κ_-1γ_-1(v)-i∑_z ∈Ω^+(κ_z(ω)γ_z(v)+κ_z(ω)γ_z(v))+η(v),   v ∈,

where Ω^+={z ∈Ω | z>0} and, as before, γ_z(v)=-iγ(zv) and η is a continuous symbol on . The same strategy used in the proof of Theorem <ref> leads to the following identity 

    _τ^±(t;(ω))   =_τ_1^±(t;κ_1(ω)(γ_1))+_τ_1^±(t;κ_-1(ω)(γ_-1))
       +∑_z ∈Ω^+_τ_1^±(t;κ_z(ω)(γ_z)+κ_z(ω)(γ_z)).

The fact that the jumps of ω are arranged symmetrically around  can be used to show that the positive and negative eigenvalues of the compact operator 
    κ_z(ω)_N(γ_z)+κ_z(ω)_N(γ_z)
 are arranged almost symmetrically around 0, in a sense that we will specify in Lemma <ref>-<ref>. Using Theorem <ref>, we conclude that 

    _τ_1^±(t;κ_z(ω)(γ_z)+κ_z(ω)(γ_z))=_τ_1(t;κ_z(ω)(γ_z)).

 Using once again the result of Widom in (<ref>), we arrive at (<ref>). It is worth noting here that (<ref>) shows that if ω has jumps occurring at a pair of complex conjugate points, then the upper and lower logarithmic spectral densities, ^±(t, (ω)), contribute equally to the logarithmic spectral density of (ω), we refer to this as the “Symmetry Principle", following the terminology used by the authors of <cit.>.

Both Theorem <ref> and <ref> are then extended to the case of a symbol with infinitely-many jump-discontinuities using an approximation argument first presented by Power in <cit.> and subsequently in <cit.>, see Section 4 below. 




Abstract properties of the spectral density
First definitions and results
Let  denote the ideal of compact operators. For any p>0,  denotes the ideal of compact operators whose singular values are p-summable and let _0=∩_p>0.  For p≥ 1, the -norm is defined as 
 
    A_^p=∑_n=1^∞s_n(A)^p,   1≤ p<∞.
    A_=sup_ns_n(A),   p=∞.

 Here {s_n(A)}_n=1^∞ is the sequence of singular values of A ordered in a decreasing manner with multiplicities taken into account. All operators in this section are bounded operators acting on the space of square summable sequence .

The functions , _± were defined in the Introduction. It is clear that (t; A)=(t; A^∗), as the non-zero singular values of A and A^∗ coincide and, furthermore one has 

    (t; A)=(t^2; A^∗A),     t>0.

For any self-adjoint operator A, the functions  and _± are linked via the following: 
    (t;A)=_+(t;A)+_-(t;A),     t>0.

The singular-value counting function of K ∈, satisfies the following simple estimate:

Let K ∈, 1≤ p< ∞, then, for any t>0 one has 
    (t; K)≤K__p^p/t^p.
 If K is self-adjoint, the same holds for the functions _±(t;K).

We will also use the following inequalities, known as Weyl's inequalitites, see <cit.>:
 Let A, B be compact operators and 0<s<t, then

    (t;A+B)≤(t-s;A)+(s;B),
    _±(t;A+B)≤_±(t-s;A)+_±(s;B),

with the last inequality holding for self-adjoint operators.  




For a bounded  τ on [0, ∞)^2 we have already defined in the Introduction the meaning of τ_N⋆ A. We have the following simple

Let τ be continuous at (0,0) with τ(0,0)=1 and suppose it satisfies Assumption <ref>. Then for any bounded operator A, τ_N⋆ A→ A as N →∞ in the strong operator topology. Furthermore, if A is compact, the same is true in the operator norm.


Recall that Assumption <ref> implies that for any operator A one has 

    τ_N⋆ A≤sup_N≥ 1τ_N_A.

Let e_j,  j≥ 0 be the standard basis vectors of . Using continuity of τ at (0,0) and the fact that τ(0,0)=1, a simple calculation shows (τ_N⋆ A)e_j→ Ae_j in  and so we obtain that (τ_N⋆ A)x→ A x as N →∞ for any finite sequence x ∈. 

For any x ∈, the result follows from a standard /3 argument. In particular, for >0, we find a finite sequence x_ so that x-x__2</3. Using the triangle inequality and (<ref>), together with the fact that (τ_N⋆ A)x_→ A x_ we obtain the assertion.

If A is compact, we have that for any given >0 we can find a finite matrix B so that A-B<. For any finite matrix B, the convergence τ_N⋆ B→ B in the strong operator topology implies convergence in the operator norm, so for N large we have (τ_N⋆ B)- B<. The triangle inequality now yields

    (τ_N⋆ A)-A   ≤τ_N⋆(A-B)+τ_N⋆ B-B+B-A
       ≤(1+sup_N≥ 1τ_N_)A-B+
       ≤(2+sup_N≥ 1τ_N_)


As a consequence, we have the following

Let K ∈ and τ be as in Lemma <ref>. Then for any t>0 one has 
    (t;τ_N⋆ K)=O_t(1),   N →∞.
 If τ(x,y)=τ(y,x) and K is self-adjoint, the same holds for the functions _±.


From Lemma <ref>, we have that τ_N⋆ K→ K in the operator norm and, in particular, for >0 we can find N suitably large so that τ_N⋆ K-K<, whereby it follows that (;τ_N⋆ K-K)=0. Using (<ref>), we obtain for 0<<t:

    (t;τ_N⋆ K)   ≤(t-;K)= C_t.

The proof in the self-adjoint case follows exactly the same reasoning.

Define _0 as the set of operators on :

    A ∈_0  ⟺   A_j,k= O(1/j+k),   j,k →∞.

Clearly A ∈_0 if and only if there exists a sequence a ∈ℓ^∞(^2) so that 
    A_j,k=a_j,k/π(j+k+1),    ∀ j, k≥ 0.
 From Hilbert inequality one obtains the estimate A≤a_ℓ^∞, and so A is also bounded. If the multiplier τ satisfies assumption <ref>, i.e. if for some α>1/2, one has 
    τ(x,y)≤C_α/log(x+y+2)^α,   ∀ x, y,
 it is not difficult to see that when A ∈_0 one has that τ_N⋆ A ∈_2, since we have the following estimate

    τ_N⋆ A__2^2   =∑_j,k,≥ 0τ(j/N,  k/N)A_j,k^2
       ≤ C_α∑_j,k≥ 01/log(j+k/N+2)^2α(j+k+1)^2<∞.

In particular, τ_N⋆ A is a compact operator for any given N and so it makes sense to study how the functions (t;τ_N⋆ A) and _±(t;τ_N⋆ A) (whenever A is self-adjoint and τ(x,y)=τ(y,x)) behave for large N. To this end, it is useful to define the following two functionals

    _τ(t; A):=lim sup_N→∞(t; τ_N⋆ A)/log(N),   t>0,
    _τ(t; A):=lim inf_N→∞(t; τ_N⋆ A)/log(N),   t>0.

If _τ(t;A)=_τ(t;A), we denote by _τ(t;A) their common value. For a self-adjoint operator A ∈_0, we define the functionals _τ^±(t;A), _τ^±(t;A) with the functions _± replacing  in (<ref>) and (<ref>) respectively and denote by _τ^±(t; A) their common value, if it exists. 


Invariance of spectral densities
For a fixed operator A ∈_0, we wish to study the relation between the asymptotic behaviour of 𝗇(t;τ_N⋆ A) for large N and the Schur-Hadamard multiplier τ. In particular, the result below tells us that the function 𝗇(t;τ_N⋆ A) (as well as 𝗇_±(t;τ_N⋆ A)) asymptotically behaves independently of the multiplier τ. We refer to this phenomenon as the Invariance Principle and we state it as follows


Suppose τ_1, τ_2 are multipliers satisfying assumptions <ref> and <ref>. Then for A ∈_0 and for t>0 one has that 

    _τ_1(t+0;A)≤_τ_2(t;A)≤_τ_1(t-0;A),_τ_1(t+0;A)≤_τ_2(t;A)≤_τ_1(t-0;A).

Similarly, for a self-adjoint A∈_0 and τ_i(x,y)=τ_i(y,x), then one has that 

    _τ_1^±(t+0;A)≤_τ_2^±(t;A)≤_τ_1^±(t-0;A),_τ_1^±(t+0;A)≤_τ_2^±(t;A)≤_τ_1^±(t-0;A).


Before proving the result, let us prove the following auxiliary lemma.

Let σ  satisfy Assumption <ref> and be such that σ(0,0)=0 and such that for some >0 and some β >1/2, there exists C_β>0, so that 

    σ(x, y)≤ C_βlog(x+y)^-β,   ∀  0≤ x, y≤.

For any A ∈_0, one has σ_N⋆ A∈_2 and furthermore there exists C>0, independent of N, such that 
    σ_N⋆ A__2≤ C.



We need to estimate the following quantity 
    σ_N⋆ A__2^2=∑_j,k≥ 0σ(j/N, k/N)^2A_j,k^2.

A modification of the integral test and the assumption that A ∈_0, shows that one can find C>0 so that

    σ_N⋆ A__2^2   ≤ C ∬_^2σ(x/N, y/N)^2/(x+y+1)^2dxdy
       =C ∬_^2σ(s, t)^2/(s+t+1/N)^2dsdt   (:=I_N),

the last inequality follows from the change of variables x=Ns,  y=Nt. Let Ω_={(s, t) ∈^2 |  s^2+t^2<} and Ω_^c=^2∖Ω_, then:

    I_N   = ∬_Ω_σ(s, t)^2/(s+t+1/N)^2dsdt      (:=J_1)
       +∬_Ω_^cσ(s, t)^2/(s+t+1/N)^2dsdt     (:=J_2)

We will show that each summand is uniformly bounded. Since σ satisfies (<ref>), it follows

    J_1   ≤C_β/log(2)^2∬_Ω_1/log(s^2+t^2)^2β(s^2+t^2)dsdt
       ≤ C ∫_0^1/rlog(r)^2βdr<∞.

The second inequality is a consequence of writing the integral in polar coordinates and, since β>1/2, the last integral is finite. Using <ref>, it follows that

    J_2   ≤ C ∬_Ω_^cdsdt/(s+t)^2log(s+t+2)^2α
       ≤ C ∫_^∞dx/xlog(x+2)^2α<∞.

We have thus obtained that I_N is uniformly bounded in N, whereby the assertion follows. 


Write A_i^(N)= (τ_i)_N⋆ A, Weyl's inequality (<ref>) states that 

    𝗇(t;A_1^(N))   =𝗇(t+s-s;A_1^(N)+A_2^(N)-A_2^(N))
       ≤𝗇(t-s;A_2^(N))+𝗇(s;A_1^(N)-A_2^(N)),

for any 0<s<t. Swapping the roles of A_1^(N) and A_2^(N) in the above, we obtain 

    𝗇(t+s;A_2^(N))-𝗇(s;A_1^(N)-A_2^(N))≤𝗇(t;A_1^(N))≤𝗇(t-s;A_2^(N))+𝗇(s;A_1^(N)-A_2^(N)),

 Lemmas <ref> and <ref> together imply that 
    𝗇(s;A_1^(N)-A_2^(N))=O_s(1)
 as N→∞, and so we obtain that 

    _τ_1(t+s;A)≤_τ_2(t;A)≤_τ_1(t-s;A),
    _τ_1(t+s;A)≤_τ_2(t;A)≤_τ_1(t-s;A).

Sending s→ 0 gives the desired inequalities. In the self-adjoint setting, the same reasoning carries through once we replace the function  with the functions _±.



Almost symmetric and almost orthogonal operators As mentioned in the Introduction, we will use the following two results which are similar, at least in spirit, to Theorems 2.2 in <cit.> and Theorem 2.7 in <cit.> and their proofs follow the same scheme. 

From now on, we make no assumptions on the uniform boundedness and smoothness of our multiplier τ and write A^(N)=τ_N⋆ A. The first of the two results discussed below is about the interactions at the level of their spectral densities between two operators. Namely, if A, B are bounded operators whose truncations A^(N), B^(N) are almost orthogonal in the sense that 
    A^(N)∗B^(N)∈,     A^(N)B^(N)∗∈,
 for some p≥ 1 uniformly in N, then each of the logarithmic spectral densities of A and B contributes independently to the logarithmic spectral density of A+B. Let us state the result as follows

Let A_i, with 1≤ i≤ L, be a family of operators such that for some p∈ [1, ∞) one has 
    sup_N≥ 1 A^(N)∗_jA_k^(N)_<∞,   sup_N≥ 1 A_j^(N)A^(N)∗_k_^p<∞,   ∀   j≠ k.
 Then, for A=∑_j=1^LA_j and for any t>0:

    ∑_j=1^L_τ(t+0; A_j)≤_τ(t; A)   ≤∑_j=1^L_τ(t-0; A_j),
    ∑_j=1^L_τ(t+0; A_j)≤_τ(t; A)   ≤∑_j=1^L_τ(t-0; A_j).

If all A_j are self-adjoint and τ(x,y)=τ(y,x), then we have 

    ∑_j=1^L_τ^±(t+0; A_j)≤_τ^±(t; A)   ≤∑_j=1^L_τ^±(t-0; A_j),
    ∑_j=1^L_τ^±(t+0; A_j)≤_τ^±(t; A)   ≤∑_j=1^L_τ^±(t-0; A_j).




We will prove only (<ref>), since (<ref>),(<ref>) and (<ref>) follow the same line of reasoning. Put =⊕_i=1^L and define the block diagonal operator _N={A_1^(N),…,A_L^(N)} such that 
    _N(f_1,…, f_L)=(A_1^(N)f_1,…, A_L^(N)f_L).
 Similarly, let ={A_1, …, A_L}. Define the operator :→ as 
    (f_1,…,f_L)=∑_j=1^Lf_j.

The operator (_N)^∗(_N) can be written as an L× L block-matrix of the form:

    [ A_1^(N)∗A_1^(N) A_1^(N)∗A_2^(N)               … A_1^(N)∗A_L^(N); A_2^(N)∗A_1^(N) A_2^(N)∗A_2^(N)               … A_2^(N)∗A_L^(N);               ⋮               ⋮               ⋱               ⋮; A_L^(N)∗A_1^(N) A_L^(N)∗A_2^(N)               … A_L^(N)∗A_L^(N) ].

Since the operator _N^∗_N is the block diagonal L× L matrix 
    [ A_1^(N)∗A_1^(N)               0               …               0;               0 A_2^(N)∗A_2^(N)               …               0;               ⋮               ⋮               ⋱               ⋮;               0               0               … A_L^(N)∗A_L^(N) ],
 
it is easy to see that the difference (_N)^∗(_N)-_N^∗_N is the L× L matrix

    _N=[               0 A_1^(N)∗A_2^(N)               … A_1^(N)∗A_L^(N); A_2^(N)∗A_1^(N)               0               … A_2^(N)∗A_L^(N);               ⋮               ⋮               ⋱               ⋮; A_L^(N)∗A_1^(N) A_L^(N)∗A_2^(N)               …               0 ].

Furthermore, since the operators A_j are so that sup_NA_j^(N)∗A_k^(N)_ is finite for all j≠ k, then 
    sup_N≥ 1_N_<∞.
 
Thus, Weyl inequality (<ref>) gives

    (t; _N)=(t^2; (_N)^∗(_N))   ≤(t^2-s; _N^∗_N)+(s; _N)
       =∑_j=1^L(t^2-s; A_j^(N)∗ A_j^(N))+(s; _N),

where in the second line we used the fact that _N^∗_N is diagonal and so 
    (t^2-s; _N^∗_N)=∑_j=1^L(t^2-s; A_j^(N)∗ A_j^(N)).
 Just as in the proof of Theorem <ref>, we swap the roles of _N and _N and, using (<ref>), we obtain 
    ∑_j=1^L(√(t^2+s); A_j^(N))-(s; _N)≤(t; _N)≤∑_j=1^L(√(t^2-s); A_j^(N))+(s; _N).

Dividing by log(N), exploiting the sub-additivity of the lim sup in conjunction with Lemma <ref> and sending s → 0, one gets  

    ∑_j=1^L_τ(t+0; A_j)≤_τ(t; )≤∑_j=1^L_τ(t-0; A_j).

Recall now that we set A=∑_j=1^LA_j. We have

    A^(N)A^(N)∗=∑_j, k=1^LA_j^(N)A_k^(N)∗,
    (_N)(_N)^∗=∑_j=1^LA_j^(N)A_j^(N)∗.

Write D_N=A^(N)A^(N)∗-(_N)(_N)^∗, then from our assumptions it follows that sup_N≥ 1D_N_<∞ and using (<ref>) in conjunction with the Weyl inequality (<ref>), we obtain

    (t; A^(N)∗)=(t^2; A^(N)A^(N)∗)   ≤(t^2-s; (_N)(_N)^∗)+(s;D_N)
       =(√(t^2-s);(_N)^∗)+(s;D_N).

Whereby we obtain that

    _τ(t; A)=_τ(t; A^∗)≥_τ(t+0; ()^∗)=_τ(t+0; ()),
    _τ(t; A)=_τ(t; A^∗)≤_τ(t-0; ()^∗)=_τ(t-0; ()).

The above, in conjunction with (<ref>), gives the result.


The second result applies to a self-adjoint operator A and establishes a relation between _τ^+(t;A) (resp. ^+(t;A)) and _τ^-(t;A) (resp. ^-(t;A)).  More precisely, if a self-adjoint operator A is so that its truncation A^(N) is almost symmetric under reflection around 0, in the sense that for some unitary operator U one has 
    UA^(N)+A^(N)U ∈
 for some p≥ 1 uniformly in N, then its upper and lower logarithmic spectral densities contribute equally to the logarithmic spectral density of A. In other words, the positive and negative eigenvalues of τ_N⋆ A accumulate to the spectrum of A in the same way. We can formulate this as follows

Let A be a self-adjoint operator and let τ be such that τ(x,y)=τ(y,x). Suppose there exists a unitary operator U for which 
    sup_N≥ 1UA^(N)+A^(N)U_<∞
 for some p≥ 1. Then for t>0

    _τ^-(t+0;A)≤_τ^+(t;A)≤_τ^-(t-0;A),
    _τ^-(t+0;A)≤_τ^+(t;A)≤_τ^-(t-0;A)

In particular, we get that
 
    _τ(t+0;A)≤ 2_τ^±(t;A)≤_τ(t-0;A),
    _τ(t+0;A)≤ 2_τ^±(t;A)≤_τ(t-0;A).





Write 
    K_N=A^(N)+U^∗A^(N)U.
 By assumption sup_N≥ 1K_N_ is finite. Furthermore, by Weyl inequality (<ref>)

    _±(t; A^(N))   =_±(t; -U^∗A^(N)U+K_N)
       ≤_±(t-s; -U^∗A^(N)U)+_±(s,K_N)
       =_∓(t-s; A^(N))+_±(s,K_N),

where 0<s<t. In particular, this gives that

    _-(t+s; A^(N))-_-(s,K_N)≤_+(t; A^(N))≤_-(t-s; A^(N))+_+(s,K_N).

The result follows once we divide through by log(N), send N →∞ and use Lemma <ref>.




Hankel operators and the Abel summation method
Hankel operators
In the Introduction, we defined Hankel matrices acting on , equivalently they can also be defined as integral operators acting on .

Let  be the unit circle in the complex plane, and  the Lebesgue measure normalised to 1, i.e d(z)=(2π iz)^-1dz. Define the Riesz projection as 

    P_+:⟶
    (P_+f)(v)=lim_→ 0∫_f(z)z/z-(1-)vd(z),   v ∈.

For a symbol ω, the Hankel operator H(ω) is:

    H(ω ):→
    H(ω)f= P_+ω JP_+f
 
where J is the involution Jf(v)=f(v) and, by a slight abuse of notation, ω denotes both the symbol and the induced  operator of multiplication on . We can immediately see that if ω satisfies (<ref>), H(ω) is self-adjoint. Furthermore, it is easy to see that

    H(ω)≤ω_∞.



For any non-negative integers j,k, one has 

    (H(ω)z^j, z^k)_   =(P_+ω JP_+z^j, z^k)_
       =(ω· z^-j, z^k)_=ω(j+k),

and so the matrix representation of H(ω) in the basis {z^n}_n∈ is the block-matrix

    [   0   0;   0 (ω) ],
  with respect to the orthogonal decomposition =H^2⊕(H^2)^⊥, where H^2 is the closed linear span in  of the monomials {z^n}_n≥ 0. In other words, H(ω) and (ω) are unitarily equivalent (modulo kernels) under the Fourier transform.

For 0<r<1, let P_r be the Poisson kernel, defined as

    P_r(v)=∑_j=-∞^∞r^nv^n=1-r^2/1-rv^2,   v∈.

For ω_r=P_r∗ω, we have the identity 

    H(ω_r)=C_rH(ω)C_r,

 where C_r is the operator of convolution by P_r on . Furthermore, it is unitarily equivalent (modulo kernels) to the Hankel matrix 

    ^(r)(ω)={r^j+kω(j+k)}_j,k≥ 0.

Note that for r=e^-1/N, the above reduces to the truncation considered in (<ref>). For 0<r<1, the map H(ω)↦ H(ω_r) has the following properties 


  * for any bounded Hankel operator H(ω), H(ω_r)∈_0. Furthermore, (<ref>) and Hölder's inequality for Schatten classes (see <cit.>) give  for 1≤ p≤∞ 
    H(ω_r)_≤1/(1-r^2p)^1/pH(ω);


  * if H(ω) ∈ for some 1≤ p≤∞, then (<ref>) implies H(ω_r)_≤H(ω)_.



Almost Orthogonal and Almost Symmetric Hankel operators
Recall that for a function  η:→,  its singular support, denoted η, is defined as the smallest closed subset, , of  such that η∈ C^∞(\). 

The following statements hold


  *  Let ω_1, ω_2 ∈ L^∞() have disjoint singular supports. Set (ω_i)_r=P_r∗ω_i, i=1,2. Then

    sup_r<1H((ω_1)_r)^∗H((ω_2)_r)__1<∞,  sup_r<1H((ω_1)_r)H((ω_2)_r)^∗__1<∞;


  *  Suppose ω∈ L^∞() be such that ± 1 ∉(ω). Let (v)=((v)),  v∈. Then 
    sup_r<1 H(ω_r)+H(ω_r)__1<∞.
 




Similar results are already known in the literature, but only from a qualitative standpoint. In fact, under the same assumptions of <ref>, it is known that both H(ω_1)^∗H(ω_2) and H(ω_2)^∗H(ω_1) ∈_0. Similarly for <ref>, it is also known that H(ω)+H(ω)∈_0. For a proof of both facts see <cit.> and <cit.> respectively, even though both facts are already mentioned in <cit.>.

To prove the statements in Lemma <ref>, we use the following:
 

  *  if K is an operator on  with integral kernel k ∈ C^∞(^2), then K ∈_1;

  *  H(ω) ∈_1 if ω∈ C^2() and furthermore there exists C>0 such that:
    H(ω)__1≤ω_∞+Cω”_2.


  *  if ω∈ C^2(), the commutator [P_+, ω] is trace-class.



<ref> is folklore. It can be proved by approximating the kernel k by trigonometric polynomials.

Let us prove <ref>. First, recall two facts:


  * any ω∈ C^2() is the uniform limit of the sequence 
    ω_N(v)=∑_j≤ N-1ω(j)v^j,   v ∈.
 Thus for any N and v ∈, the Cauchy-Schwarz inequality together with Plancherel Identity give:

    |ω(v)-ω_N(v)|   ≤ω”_2(2∑_j≥ Nj^-4)^1/2≤ C  ω”_2 N^-3/2.


  * for A ∈ one has s_N(A)=inf{A-B  | rank(B)≤ N-1}, N≥ 1, and, in particular, s_1(A)=A.

Putting these two facts together and noting that rank(H(ω_N))≤ N, we have that for N≥ 2:

    s_N(H(ω))   =inf{H(ω)-B | rank(B)≤ N-1}
       ≤H(ω)-H(ω_N-1)
       ≤ω-ω_N-1_∞
       ≤ Cω”_2/(N-1)^3/2.

Thus we see that H(ω) ∈_1 and, furthermore, 

    H(ω)__1=s_1(H(ω))+∑_n≥ 2s_n(H(ω))≤ω_∞+C ω”_2.

<ref>. Write P_-=I-P_+, where I is the identity operator. Since P_+ is a projection and P_+P_-=P_-P_+=0, one has 
    [P_+, ω]=[P_+,(P_++P_-) ω](P_++P_-)=P_+ω P_--P_-ω P_+.
 Using the identity P_-=JP_+J-P_+JP_+, it follows that 
    [P_+, ω]=H(ω)J-JH(ω)^∗-P_+ω P_+JP_++P_+JP_+ω P_+.

Since P_+JP_+ is a rank-one operator (projection onto constants), [P_+, ω] is trace-class if and only if H(ω)J-JH(ω)^∗ is, which follows immediately from <ref>.

With these facts at hand, we are now ready to prove Lemma<ref>.

<ref>: we will only show the first inequality, as the second can be proved in the same way.  From the assumptions on ω_1, ω_2, we can find ζ_1, ζ_2 ∈ C^∞() such that ζ_1∩ζ_2=∅ and such that (1-ζ_i)ω_i vanishes identically in a neighbourhood of ω_i. We will repeatedly use the following two facts:


  *  for any ϕ∈ L^∞(), Young's inequality holds, i.e one has the estimate:

    P_r∗ϕ_∞≤ϕ_∞;


  *  one has that P_r∗ω∈ C^∞() and furthermore (P_r∗ω) →ω as r→ 1- locally uniformly on ∖ω. The same is true for its derivatives (P_r∗ω)^(n). 


We set ζ_i=1-ζ_i,  i=1,2 and use the triangle inequality to obtain

    H((ω_1)_r)^∗H((ω_2)_r)__1   ≤H(ζ_1(ω_1)_r)^∗H(ζ_2(ω_2)_r)__1
    														 +H(ζ_1(ω_1)_r)^∗H(ζ_2(ω_2)_r)__1
       +H(ζ_1(ω_1)_r)^∗H(ζ_2(ω_2)_r)__1
    														 +H(ζ_1(ω_1)_r)^∗H(ζ_2(ω_2)_r)__1,

from which we see that it is sufficient to find uniform bounds for each summand above.

Recall that H(ω_i)=P_+ω_iJP_+ and P_+ is a projection, thus: 
    H(ζ_1(ω_1)_r)^∗H(ζ_2(ω_2)_r)=P_+Jζ_1(ω_1)_rP_+ζ_2(ω_2)_rJP_+.
 Since ζ_1 and ζ_2 have disjoint supports, the operator ζ_1P_+ζ_2 has a C^∞(^2) integral kernel given by 
    ζ_1(z)ζ_2(v)/v-zv,   v,  z ∈ .
 Lemma <ref>-<ref> shows that ζ_1P_+ζ_2∈_1. Furthermore, using Hölder inequality for the Schatten classes and (<ref>), we deduce that

    sup_r<1H(ζ_1(ω_1)_r)^∗H(ζ_2(ω_2)_r)__1≤ω_1_∞ω_2_∞ζ_1P_+ζ_2__1<∞.
 

By Lemma <ref>-<ref>, we also have that H(ζ_1(ω_1)_r) ∈_1 and furthermore: 

    H(ζ_1(ω_1)_r)^∗H(ζ_2(ω_2)_r)__1   ≤H(ζ_1(ω_1)_r)__1H(ζ_2(ω_2)_r)
       ≤ζ_2_∞ω_2_∞(C(ζ_1(ω_1)_r)”_2+ω_1_∞),

for some C>0 independent of r. In (<ref>) we used once more the Hölder inequality for Schatten classes together with the estimates (<ref>) and (<ref>). 

From <ref> and the fact that ζ_iω_i vanishes identically on ζ_i, we conclude that (ζ_i(ω_i)_r)”→ (ζ_iω_i)” uniformly on the whole of , and so

    sup_r<1(ζ_i(ω_i)_r)”_2<∞.

Using (<ref>) in (<ref>) finally gives 
    sup_r<1H(ζ_1(ω_1)_r)^∗H(ζ_2(ω_2)_r)__1<∞.

Similarly one can show that

    sup_r<1H(ζ_1(ω_1)_r)^∗H(ζ_2(ω_2)_r)__1<∞,  sup_r<1H(ζ_1(ω_1)_r)^∗H(ζ_2(ω_2)_r)__1<∞.

<ref> Since ± 1 ∉ω, then we can write ω=ϕ+η for some η∈ C^∞() and some ϕ vanishing identically in a neighbourhood U of ± 1. With this decomposition of ω, we can see that 
    H(ω_r)=H(ϕ_r)+H(η_r).
 Since η is smooth, then H(η) ∈_1 and so the triangle inequality and Hölder inequality for Schatten classes imply that 
    sup_r<1 H(ω_ r)+H(ω_r)__1≤ 2 H(η)__1+sup_r<1 H(ϕ_ r)+H(ϕ_r)__1.
 So it is sufficient to consider those symbols ω vanishing on a neighbourhood, U, of ± 1.

Fix a smooth function ζ such that 0≤ζ≤ 1,  it vanishes identically on some open V⊂ U so that ± 1 ∈ V, ζ≡ 1 on \ U and such that ζ(v)=ζ(v),  v∈. We can write:

    H(ω_r)+H(ω_r)   = H((1-ζ)ω_r)+H((1-ζ)ω_r)
       + H(ζω_r)+H(ζω_r)

Let us study these operators  more closely.
Using the triangle inequality, we obtain that 

    sup_r<1 H((1-ζ)ω_r)+H((1-ζ)ω_r)__1≤ 2sup_r<1H((1-ζ)ω_r)__1.

Using <ref> and the fact that (1-ζ)ω≡ 0 on , we conclude that ((1-ζ)ω_r)”→ 0 on  and so Lemma <ref>-<ref> gives

    sup_r<1H((1-ζ)ω_r)__1≤sup_r<1(ω_∞+C((1-ζ)ω_r)”_2)<∞.

For the operators appearing in the second line of (<ref>), write 
    H(ζω_r)+H(ζω_r)=([, P_+]ζ)ω_rJP_++P_+ω_rJ(ζ[P_+, ]).
 Let us now prove that the commutators [, P_+]ζ, ζ[, P_+] ∈_1. By our choice of  and ζ, we have J=- J and Jζ=ζ J, whence it follows that

    [, P_+]ζ   = P_+ζ-ζ P_++ζ P_+-P_+ζ=[P_+, ζ]+[ζ, P_+],
    ζ[, P_+]   =ζ P_+-P_+ζ+P_+ζ -ζ P_+=[ζ, P_+]+[P_+, ζ].

Furthermore, our choice of ζ gives that the product ζ∈ C^∞(), and Lemma <ref>-<ref> together with (<ref>) implies that

    sup_r<1 H(ζω_r)+H(ζω_r)__1≤ω_∞([, P_+]ζ__1+ζ[P_+, ]__1)<∞.

Putting together (<ref>), (<ref>) and (<ref>) and using the triangle inequality on (<ref>) gives the assertion.





Spectral density of our model operator: the Hilbert matrix
An important ingredient to the proof of all our results is the model operator for which it is possible to explicitly compute the spectral density. Following the ideas of previous works, <cit.>, a natural candidate is the Hilbert matrix, given by the symbol γ defined in (<ref>). Putting together the result of Widom, see <cit.> and the Invariance Principle <ref>, we obtain

 Let τ satisfy assumptions <ref> and <ref>, then one has that 
    _τ(t;(γ))=_τ(t;(γ))=𝖼(t),   t>0,
 where 𝖼 has been defined in (<ref>). If τ(x,y)=τ(y,x), then we also have 
    ^+_τ(t;(γ))=^+_τ(t;(γ))=𝖼(t),
    ^-_τ(t;(γ))=^-_τ(t;(γ))=0.


As an immediate consequence of the above, we obtain

Let z ∈ be fixed and let γ_z(v)=-iγ(zv). Then the same result of Proposition <ref> holds for the operator (γ_z).

 The Invariance Principle, Theorem <ref>, shows that it is sufficient for the statement to hold for τ_□(x, y)=_□(x,y) defined in (<ref>). This has already been done in <cit.> and it has already been discussed in the Introduction in (<ref>). Since the Hilbert matrix is a positive-definite operator, it is easy to see that τ_N⋆(γ) is positive-definite and so 

    _τ(t;(γ))=^+_τ(t;(γ)),   ^-_τ(t;(γ))=0.

The statement can be independently proved using the function τ_1(x,y)=e^-(x+y) discussed in the Introduction, however we postpone this to the Appendix.


Indeed, note that γ_z(j)=-i z^j γ(j),  j≥ 0. Hence, for any function τ one has: 
    τ_N⋆(γ_z)=-iU_z(τ_N⋆(γ))U_z
 where U_z is the unitary operator of multiplication by  z^j,  j≥ 0. From this, we immediately see that 
    s_n(τ_N⋆(ψ_z))=s_n(τ_N⋆(γ)),  ∀ n≥ 1
 and so the statement follows immediately from Proposition <ref>.




Proof of Theorem <ref>
The proof of the result will be broken down in two Steps. For brevity, we denote by ^(N)(ω) the operator τ_N⋆(ω). We also recall that Ω is the set of jump-discontinuities of the symbol ω and  is the function in (<ref>).

Step 1. Finitely many jumps. Suppose that Ω is finite. Setting γ_z(v)=-iγ(zv), with γ being the symbol defined in (<ref>), write

    ω(v)=∑_z ∈Ωκ_z(ω)γ_z(v)+η(v)

 where η is continuous on  and let Φ denote the symbol 
    Φ(v)=∑_z ∈Ωκ_z(ω)γ_z(v).
 Weyl's inequality (<ref>) shows that for 0<s<t one has

    (t+s;^(N)(Φ))-(s;^(N)(η))≤(t;^(N)(ω))≤(t-s;^(N)(Φ))+(s;^(N)(η)).

Since (η) is compact, Lemma <ref> shows that (s;^(N)(η))=O_s(1) as N→∞ and so, using the definition of the functionals _τ, _τ we deduce that for any t>0

    _τ(t; (ω))≤_τ(t-0; (Φ)), 
    _τ(t; (ω))≥_τ(t+0; (Φ)).
 
Integration by parts shows that 

    Φ(j)   =∑_z ∈Ωκ_z(ω)γ_z(j)
       =-i/π (j+1)∑_z ∈Ωκ_z(ω)z^j=O(1/j+1),   j→∞

and so by the Invariance Principle <ref> applied to the operator (Φ), we obtain

    _τ(t; (Φ))≤_τ_1(t-0; (Φ)), 
    _τ(t; (Φ))≥_τ_1(t+0; (Φ)).
 
where τ_1(x,y)=e^-(x+y) induces the regularisation in (<ref>). The Fourier transform  on , defined as 
    ( f)(j)=∫_f(z)z^jd(z),   f ∈,  j≥ 0,
 implies that modulo kernels, see (<ref>), we have

    ^(N)(Φ)=∑_z∈Ωκ_z(ω)^(N)(γ_z)=∑_z ∈Ωκ_z(ω) H((γ_z)_N)^∗,

where (γ_z)_N=P_r∗γ_z, with P_r being the Poisson kernel with r=e^-1/N. By Lemma <ref>-<ref> and unitary equivalence, we have that whenever z≠ w

    sup_N≥ 1^(N)(γ_z)^∗^(N)(γ_w)__1=sup_N≥ 1H((γ_z)_N)^∗H(γ_w)_N__1<∞.

Using Theorem <ref>, it then follows that for t>0

    _τ_1(t; (Φ))≤∑_z ∈Ω_τ_1(t-0; κ_z(ω)(γ_z)), 
    _τ_1(t; (Φ))≥∑_z∈Ω_τ_1(t+0; κ_z(ω)(γ_z)).
 
Finally, Corollary <ref> together with (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) and the continuity of  at t≠ 0 gives that

    _τ(t;(ω))≤∑_z ∈Ω𝖼(t/κ_z(ω)),_τ(t;(ω))≥∑_z∈Ω𝖼(t/κ_z(ω)).

The obvious inequality _τ(t;H(ω))≤_τ(t;H(ω)) proves the assertion.

We note that (<ref>) and (<ref>) hold if we consider any symbol ω which is smooth except for a finite set of jumps discontinuities. These two together are yet another instance of the Localisation principle we referred to in the Introduction.

Step 2. From finitely many to infinitely jumps. Suppose now that Ω is infinite. Define the sets:

    Ω_0={z ∈ | κ_z(ω)≥ 2^-1},
    Ω_n={z ∈ | 2^-n-1≤κ_z(ω)< 2^-n},   n≥ 1.

As we mentioned earlier, these are finite. Let ϕ_n be functions such that ϕ_n=Ω_n, κ_z(ϕ_n)=κ_z(ω) for any z ∈Ω_n and such that 
    ϕ_n_∞=max_z ∈Ω_nκ_z(ω).

Let Φ=∑_n≥ 0ϕ_n∈ L^∞(). Since ω - Φ∈ C(), the operator (ω)-(Φ) ∈ and so, by Lemma <ref> once again we obtain

    _τ(t; (ω))≤_τ(t-0; (Φ)),
    _τ(t; (ω))≥_τ(t+0; (Φ)).

For a fixed s>0, let M be so that Φ-Φ_M_∞< s, where Φ_M=∑_n=0^Mϕ_n. The uniform boundedness of τ then gives 
    τ_N⋆((Φ)-(Φ_M))≤(sup_N≥ 1τ_N_)Φ-Φ_M_∞< (sup_N≥ 1τ_N_) s:=s'.
 Letting Ω_M=∪_n=0^MΩ_n, we then obtain that:

    _τ(t; H(ω))≤_τ(t-s'; H(Φ_M))=∑_z∈Ω_M(t-s'/κ_z(ω)),
    _τ(t; H(ω))≥_τ(t+s'; H(Φ_M))=∑_z∈Ω_M(t+s'/κ_z(ω)).

The equalities above follow from the Step 1., since Φ_M has finitely many jumps. Finally, sending s → 0 and noting that there are only finitely many z ∈Ω such that t≤κ_z(ω), one obtains 

    _τ(t; H(ω))=_τ(t; H(ω)).



Proof of Theorem <ref>
Just as in the proof of Theorem <ref>, we break the argument into two steps, and use the same notation as before for the operator τ_N⋆(ω) and for the symbols γ_z. We also set Ω^+={z ∈Ω |  z>0}.

Step 1. Finitely many jumps. Just as before, suppose that the symbol ω has finitely-many jump-discontinuities. Write

    ω(v)=(κ_1(ω)γ_1(v)+κ_-1(ω)γ_-1(v)+∑_z ∈Ω^+κ_z(ω)γ_z(v)+κ_z(ω)γ_z(v))+η(v),

where η is continuous on . If ω has no jump at ± 1, the corresponding quantities do not appear in the above. Denoting by Φ the sum in the brackets, Weyl inequality (<ref>) gives for 0<s<t

    _±(t+s;^(N)(Φ))-(s;^(N)(η))≤_±(t;^(N)(ω))≤_±(t-s;^(N)(Φ))+_±(s;^(N)(η)).

By Lemma <ref>, we obtain that _±(s;^(N)(η))=O_s(1), and so, for any t>0, it follows that

    _τ^±(t; (ω))≤_τ^±(t-0; (Φ)), 
    _τ^±(t; (ω))≥_τ^±(t+0; (Φ)).
 
Integration by parts once again shows that 

    Φ(j)=O(1/j+1),   j→∞.

Applying Theorem <ref> to (Φ), we see that it is sufficient to prove the result for the multiplier τ(x,y)=e^-(x+y). Since the symbols 
    κ_1(ω)γ_1, κ_-1(ω)γ_-1, κ_z(ω)γ_z+κ_z(ω)γ_z
 have mutually disjoint singular supports for z ∈Ω^+, Lemma <ref>-<ref> and Theorem <ref> imply that for t>0

    ^±_τ(t; (Φ))   ≤^±_τ(t-0; κ_1(ω)(γ_1))+
    ^±_τ(t-0; κ_1(ω)(γ_1))
       +∑_z ∈Ω^+^±_τ(t-0; κ_z(ω)(γ_z)+κ_z(ω)(γ_z)) 
    ^±_τ(t; (Φ))   ≥^±_τ(t+0; κ_1(ω)(γ_1))+
    ^±_τ(t+0; κ_1(ω)(γ_1))
       +∑_z ∈Ω^+^±_τ(t+0; κ_z(ω)(γ_z)+κ_z(ω)(γ_z)) .

The operators κ_± 1(ω)(γ_± 1) are sign definite, and furthermore one has that 

    κ_± 1(ω)(γ_± 1)≥ 0 (resp.≤ 0) if -i κ_± 1(ω)≥ 0 (resp.≤ 0).

In either case, Proposition <ref> gives that

    ^±_τ(t;κ_± 1(ω)(γ_± 1))   =_±(-i κ_± 1(ω))_τ(t;κ_± 1(ω)(γ_± 1))
       =_±(-i κ_± 1(ω))𝖼(tκ_± 1(ω)^-1)

where _± is the indicator function of _±=(0, ±∞).

From Lemma <ref>-<ref>, Theorem <ref> and Theorem <ref> above, we get that for any z ∈Ω^+

    ^±_τ(t; κ_z(ω)(γ_z)+κ_z(ω)(γ_z))   =1/2_τ(t; κ_z(ω)(γ_z)+κ_z(ω)(γ_z))
       =𝖼(tκ_z(ω)^-1).

Using (<ref>) and (<ref>) in (<ref>) and (<ref>), the continuity of 𝖼 at t≠ 0 gives that 

    ^±_τ(t; (ω))=^±_τ(t; (Φ))

and so we arrive at (<ref>).


As we wrote earlier in the Introduction, if the symbol has a pair of complex conjugate jumps, then (<ref>) shows that the upper and lower logarithmic spectral density of (ω) contribute equally to the logarithmic spectral density of (ω). This is  an effect of the Symmetry Principle we referred to in the Introduction.


Step 2. From finitely many to infinitely many jumps. For fixed s > 0, define the set 
    Ω_s^+={z ∈Ω | κ_z(ω)>s  and  z> 0}.
 Just as in Step 2. in the proof of Theorem <ref>, we can find a symbol ω_s∈ PC() so that ω-ω_s_∞< s, the set of its discontinuities is precisely Ω_s^+∪{± 1} and 
    κ_z(ω)=κ_z(ω_s),  ∀ z ∈Ω_s^+∪{± 1}.
 The set Ω_s^+∪{± 1} is finite, thus from Weyl inequality (<ref>) and Step 1. we obtain

    ^±_τ(t; (ω))   ≤^±_τ(t-s'; (ω_s))
    ^±_τ(t; (ω))   ≥^±_τ(t+s'; (ω_s)),

where s'=(sup_N≥ 1τ_N_) s. Finally, sending s → 0 and using the continuity of 𝖼 at t≠ 0 establishes the result in its generality. 


The same reasoning of Step 1. in both proofs above applies in this case, with only one minor change. Since we assume that τ induces a uniformly bounded multiplier on ,  p>1, i.e. that (<ref>) holds, in (<ref>) and (<ref>) we need to assume that η is a symbol so that (η) ∈. Then Lemma <ref> shows that (s;^(N)(η))=O_s(1) and, in the self-adjoint case _±(s;^(N)(η))=O_s(1). The rest follows immediately.



Exactly the same reasoning of the proofs of Theorems <ref> and <ref> above applies in this case, with the only difference being that in this case τ is no longer inducing a uniformly bounded multiplier on the whole space of bounded operators, just on Hankel matrices. However, all of the terms appearing in the arguments just presented are bounded Hankel operators and so the same arguments apply in this case.






§ AN INDEPENDENT PROOF OF PROPOSITION <REF>

By virtue of Theorem <ref>, choose the function τ_1(x,y)=e^-(x+y), which yields 
    ((τ_1)_N⋆(ω))_j,k=e^-j+k/Nω(j+k)=^(r)(ω)_j,k,   r=e^-1/N,
 where ^(r)(ω) is the Poisson truncation in (<ref>). We start our proof with the following Lemma, similar to <cit.>:

 For any m ∈ one has that: 
    ^(r)(γ)^m=log(1-r)/2π∫_(1/rcosh(πη))^mdη+o(log(1-r)),   r→ 1^-.

 
 
  
Let us define the operator L:→ as follows

    (L f)(j)   =1/√(π)∫_0^1f(s)s^jds,   j≥ 0.

 Its boundedness can be established using the Schur test. A simple calculation yields the identity (γ)=L L^∗, from which if follows that, with ^(r)(γ)=(γ_r) 

    ^(r)(γ)   =1/rL_rL^∗
       =1/r(L_r)(L_r)^∗

where _r is the characteristic function of the interval (0,r) and so one obtains 
    r^m^(r)(γ)^m=(_rL^∗L_r)^m,

therefore we only need to compute the latter trace. Recall now that for any bounded operator X, there is a unitary equivalence between XX^∗|_(XX^∗)^⊥ and X^∗X|_(X^∗X)^⊥. Hence, the trace of (_rL^∗L_r)^m and that of (_rL^∗L_r)^m coincide. Note however that the operator L^∗L is an operator acting on  whose integral kernel is: 
    k(t,s)=1/π(1-ts),   t, s ∈ (0,1).
 Following the procedure described in <cit.>, define the unitary transformation: 

    U   :→ L^2()
    
    (U f)(x)   =1/cosh(x)f(tanh(x)),   x>0.

Then we have B=U L^∗LU^∗:L^2()→ L^2() is the convolution operator

    (Bf)(x)   =∫_f(y)/πcosh(x-y)dy,   x>0.

In this way, we have reduced our problem to evaluating the trace of the integral operator (_rB_r)^m , where _r is the characteristic function of the interval (0, arctanh(r)). By adding 0 to its spectrum, we also consider _rB_r as an integral operator acting on L^2(), with integral kernel 
    _r(s)_r(t)/πcosh(s-t),   s,t ∈.
 We now use the following result:

Let P be an orthogonal projection and B be a bounded operator such that PB ∈_2. Let ϕ be such that ϕ(0)=0 and ϕ”∈ L^∞(B), then:

    ϕ(PBP)- Pϕ(B)P≤ϕ”_L^∞(B)PB(I-P)__2^2.


Note that _rL^∗L ∈_2 for any r<1 so by unitary equivalence _rB ∈_2. Furthermore, the operator B is unitarily equivalent, under the Fourier Transform, to the operator of multiplication on L^2() by the function 
    1/cosh(πξ/2),   ξ∈.

Whence we can estimate (_rB_r)^m by:

    _rB^m_r   =1/2π∫__r(x)dx∫_(1/cosh(πξ/2))^mdξ
       =arctanh(r)/π∫_(1/cosh(πξ))^m dξ
       =log(1-r)/2π∫_(1/cosh(πξ))^mdξ+o(log(1-r)),   r → 1^-.

We also have that:

    _rB(1-_r)__2^2   =(_rB-B_r)(1-_r)__2^2≤[_r,B]__2^2,

thus we need to find an estimate for the Hilbert-Schmidt norm of the integral operator [_r,B], which has integral kernel given by: 

    k(t,s)=_r(t)-_r(s)/πcosh(π(t-s)),   t, s∈.

It follows that

    [_r,B]__2^2=∬_^2k^2(t,s)dtds=∫_ϕ(z)/π^2cosh^2(π z)dz,

with

    ϕ(z)   =∫_(_r(z+y)-_r(y))^2dy
       = 2 min{z, arctanhr}≤ 2z.

Whereby obtaining that

    [_r,B]__2^2≤ C∫_z/cosh^2(z)dz<∞.

Using (<ref>) and (<ref>), we obtain: 
    (^(r)(γ))^m=log(1-r)/2π r^m∫_(1/rcosh(πξ))^mdξ+o(log(1-r)),
 as r→ 1^-.

Proposition <ref> now follows from a two-step approximation argument. In the first stage, using the Weierstarss Approximation theorem and the Lemma <ref>, we prove that for any function ϕ∈ C^∞_c() one has that 

    lim_r→ 1^-ϕ(^(r)(γ))/log(1-r)=1/2π∫_ϕ(1/cosh(πη))dη.

In the second, we set r=e^-1/N and we note that we can replace log(1-e^-1/N) with log(N) in the limits above and that we can write 
    (t; ^(N)(γ))=_(t, 1)(^(N)(γ)).
 Choose sequences ϕ_n^±∈ C^∞_c() for which we have 

    0≤ϕ^-_n(x)≤_(t, 1)(x)≤ϕ^+_n(x)≤ 1,   ∀ x,

and  ϕ_n^±(x)→_(t,1)(x) pointwise in x as n →∞. From the properties of  and (<ref>) it follows
 
    _τ(t; (γ))   ≤1/2π∫_ϕ_n^+(1/cosh(πη))dη,
    _τ(t; (γ))   ≥1/2π∫_ϕ_n^-(1/cosh(πη))dη.

 Finally, an application of the Dominated Convergence Theorem gives the result.



§ ACKNOWLEDGEMENT
The content of this paper forms part of the author's PhD thesis, completed at King's College London under the supervision of Prof. A.B. Pushnitski. The author is also grateful to both Dr. M. Gebert and Dr. N. Miheisi for the numerous helpful discussions and numerous remarks on earlier versions of this paper.

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