On Convergence of Toeplitz Quantization of the Sphere

Abstract

In this paper, we give an explicit expression of the Toeplitz quantization of a $C^{\infty }$ smooth function on the sphere and show that the sequence of spectra of Toeplitz quantization of the function determines its decreasing rearrangement. We also use Toeplitz quantization to prove a version of Szegö’s Theorem.

1. Introduction

In [1,2], the authors have shown similarities between the group of area-preserving diffeomorphisms of the annulus and $SU\left(n\right)$. This work aims to make some of these analogies at the level set of the Lie algebras of these groups more rigorous. Much earlier, J. Hoppe had already shown in [3] that the structure constants of the Lie algebra of the group of volume-preserving differentiable transformations of the two-sphere are equal to the $N\to \infty$ limit of the structure constants of $SU\left(N\right)$. Li and Turki, etc., studied the Eucledean hypersurfaces isometric to spheres [4]. For a compact Kähler manifold $(M,\omega )$, we denote by ${\mathcal{D}}_M^s$ the the group of s-sobolev symplectomorphisms of M. It is proved in [5] that ${\mathcal{D}}_M^s$ is a Lie group. Its Lie algebra, denoted $S{\mathcal{D}}_M^s$, is the algebra of divergence-free vector fields. These vector fields are Hamiltonian. The Poisson algebra $\mathcal{P}\left(M\right)$ of a real valued $C^{\infty }$-function is a trivial central extension of $S{\mathcal{D}}_M^s$. In the case of the global holomorphic sections of the polarization line bundle on a Hodge manifold, it was shown that there exists a self-adjoint operator that approximates the Laplace operator on functions when composed with the Berezin–Toeplitz quantization map and its adjoint, up to a point where the error tends to zero when the power of the polarization line bundle is increased and it was proven in [6].

The Lie algebra of $SU\left(n\right)$ consists of skew-Hermitian matrices that we identify with Hermitian matrices via multiplication by $\sqrt{-1}$.

At this stage, we need to see what properties of Hermitian matrices have a counterpart in $C^{\infty }$ functions. To link the Lie algebra $su\left(n\right)$ to the Lie algebra $\mathcal{P}\left(M\right)$, Toeplitz quantization is particularly convenient for our purpose. Indeed, for $f\in \mathcal{P}\left(M\right)$, the Toeplitz quantization of f is a map $T_f^m$ from ${\Gamma }_{hol}^m$, the finite dimensional space of holomorphic sections of the $m^{th}$ tensor power of the hyperplane bundle to itself that associates to a $C^{\infty }$ function f a Hermitian matrix $T_f^m$.

Using the Toeplitz quantization procedure, it was shown in [7,8,9,10,11] that $\mathcal{P}\left(M\right)$ is a $u\left(m\right)$-quasilimit($m\to \infty )$.

In the case of the sphere, the identification of the two Lie algebra is more explicit. Let $f(r,\theta )$ be a $C^{\infty }$ function on the sphere written in the patch $[1,z=r{e}^{i\theta }].$

• We define the transpose of the function $f(r,\theta )$ to be the function $f(r,-\theta )$ because the transpose of $T_{f(r,\theta )}^m=T_{f(r,-\theta )}^m$.
• Functions that depend only on the variable r are called diagonal functions because in this case $T_f^m$ are diagonal matrices for every m.
• To define the eigenvalues of a function, let us recall that for an $m\times m$ Hermitian matrix A, the eigenvalues are solutions of the moments problem: find real numbers $x_1,x_2,\cdots ,x_m$ such that

  $$tr\left(A^p\right)=\sum _{i=1}^{i=m}{x}_i^p.$$

  (1)
  Indeed, to find the eigenvalue of the matrix A, we have to find the roots of the characteristic polynomial $det(A-\lambda I)$. But the coefficients of the characteristic polynomials are the elementary symmetric polynomials, and from Newton’s formula, the sum of a power of the roots of a polynomial can be obtained from the elementary symmetric polynomials of its roots. Therefore, the eigenvalues of the matrix are the solution of the moments problem (1).
  Also the decreasing rearrangement of a function $f:(X,\mu )\to \mathbb{R}$ is a solution to the moments problem:
  Find a real function $g:[0,1]\to \mathbb{R}$ such that

  $${\int }_X{f}^{p}d\mu ={\int }_0^1{g}^p\left(t\right)dt,\forall p\in \mathbb{N}.$$
  So, the decreasing rearrangement of the function f plays the role of eigenvalues for matrices.
• We also define the trace of the function f that we denote $tr\left(f\right)$ to be just ${\int }_{\mathbb{CP}\left(1\right)}fd\Omega$ because ${\int }_{\mathbb{CP}\left(1\right)}fd\Omega ={\displaystyle \frac{tr\left(T_f^m\right)}{m+1}}$.

Our major contributions are:

• We give a direct proof of $\mathcal{P}\left(\mathbb{CP}\right(1\left)\right)$ as a $u\left(m\right)$-quasi-limit ($m\to \infty )$.
• The eigenvalues of the Toeplitz quantization of the function determine its decreasing rearrangement.
• We use Toeplitz quantization to prove Szegö’s Theorem.

To clarify things, we purposely exposed explicit calculations for the sphere case.

Using Laplace approximation, we were able to give an asymptotic approximation of the entries of Toeplitz quantization of order m.

There are four sections in this paper. In the second section, we define Toeplitz quantization and show that it leads to a notion of classical limit. In the third section, we review the notion of decreasing rearrangement and prove that the eigenvalues of the matrix $T_f^m$ determine the decreasing rearrangement of f. In the last section, we generalize the result and prove that the spectrum of the Toeplitz quantization function on the Kahler manifold converges to the decreasing rearrangement of that function.

2. Toeplitz Quantization

Let $\omega$ be the Fubini–Study symplectic form on the sphere $\mathbb{CP}\left(1\right)$, which is in the local coordinate $[1,w]$ given by

$$\omega ={\displaystyle \frac{i}{{(1+w\overline{w})}^2}}dw\wedge d\overline{w},$$

and let $\Omega ={\displaystyle \frac{\omega }{2\pi }}$ be the volume form. The tensor power ${\mathbf{L}}^{\otimes m}$ of the standard hyperplane bundle $\mathbf{L}$ has $(m+1)$ linearly independent sections, which in local coordinate w are just $1,w,\cdots ,w^m$. The bundle ${\mathbf{L}}^{\otimes m}$ comes equipped with the Hermitian metric

$${\displaystyle h^m\left(w\right)(s_1,s_2)={\displaystyle \frac{1}{{(1+|w|}^2{)}^m}}{s}_1\left(w\right)\overline{s_2\left(w\right)}.}$$

(2)

In the sequel, it is more convenient to work with polar coordinates and for that matter the volume form is $\Omega ={\displaystyle \frac{rdrd\theta }{{(1+r^2)}^2}}$.

The scalar product associated with the metric (2) is

$${\displaystyle \langle s_1,s_2\rangle ={\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{CP}\left(1\right)}{\displaystyle \frac{1}{{(1+|w|}^2{)}^m}}{s}_1\left(w\right)\overline{s_2\left(w\right)}d\Omega .}$$

(3)

For more on holomorphic sections of line bundles, one can consult [12].

Now, let ${\Gamma }_2^m$ be the space of square-integrable sections of ${\mathbf{L}}^{\otimes m}$:

$${\Gamma }_2^m=\{s\in {\mathbf{L}}^{\otimes m},{\int }_{\mathbb{CP}\left(1\right)}{h}^m(s,s)d\Omega <\infty \}.$$

The space of holomorphic sections (the span of $1,w,\cdots ,w^m$), denoted ${\Gamma }_{hol}^m$, is a closed subspace of ${\Gamma }_2^m$. We denote the orthogonal projection ${\Gamma }_2^m\to {\Gamma }_{hol}^m$ by ${\Pi }^m$. Each function $f\in C^{\infty }\left(\mathbb{CP}\left(1\right)\right)$ gives rise to an operator $M_f$ that acts as follows

$$M_{f}s\left(p\right)=f\left(p\right)s_p,p\in \mathbb{CP}\left(1\right),s\in {\Gamma }_2^m.$$

We have, using the scalar product (3)

$$\langle M_{f}s,M_{f}s\rangle ={\int }_{\mathbb{CP}\left(1\right)}f{\left(z\right)}^2{h}^m(s,s)d\Omega \le {\parallel f\parallel }_{\infty }{\parallel s\parallel }_2.$$

which shows that $M_f$ is a bounded operator. The Toeplitz operator $T^m$ is the composition of the map $M:C^{\infty }⟶\mathcal{B}\left({\Gamma }_2^m\right)$ and the orthogonal projection ${\Pi }^m$.

Definition 1.

Let $f:\mathbb{CP}\left(1\right)\to \mathbb{R}$ be a square-integrable function. The Toeplitz quantization of f is the map $T_f^m:{\Gamma }_{hol}^m\to {\Gamma }_{hol}^m$ defined by

$$T_f^m={\Pi }^m\circ {\mathbf{M}}_f,$$

where ${\mathbf{M}}_f$ is multiplication by $f.$

Toeplitz quantization is a generalization of the familiar Toeplitz operator on the circle. Let $S^1$ be the circle, with the standard Lebesgue measure ${\displaystyle \frac{d\theta }{2\pi }}$, and $L^2\left(S^1\right)$ be the Hilbert space of square-integrable functions. A bounded measurable function g on $S^1$ defines a multiplication operator $M_g$ on $L^2\left(S^1\right)$. Let $L^m$ be the subspace of $L^2\left(S^1\right)$ spanned by the functions $e^{ik\theta },0\le k\le m$. Let ${\Pi }^m$ be the orthogonal projection from $L^2\left(S^1\right)$ onto the space $L^m$. The Toeplitz operator with symbol g is defined by:

$$T_g^m:={\Pi }^m{M}_g{\Pi }^m.$$

The matrix of $T_g^m$ has the form of a Toeplitz matrix

$$\left(\begin{array}{ccccc}{a}_0& a_1& a_2& \cdots & a_m\\ a_{-1}& a_0& a_1& \cdots & a_{m-1}\\ ⋮& \ddots & \ddots & a_0& a_1\\ a_{-m}& a_{-m+1}& \cdots & a_{-1}& a_0\end{array}\right)$$

where $g={\sum }_{-\infty }^{+\infty }{a}_n{e}^{in\theta }$.

The following proposition shows that asymptotically, the matrix $T_f^m$ looks like a Toeplitz matrix.

Proposition 1.

With the usual coordinates $w=r{e}^{i\theta }$, we have:

1. 

Set ${\gamma }_{m,k}=\sqrt{m+1}\sqrt{\left({\displaystyle \genfrac{}{}{0pt}{}{m}{k}}\right)}$.

The sections ${\gamma }_{m,k}{w}^k,k=0,\cdots ,m$ constitute an orthonormal basis of of ${\Gamma }_{hol}^m$.

2. 

$T_f^m$ is a Hermitian matrix and its $(k,l)$ entry is

$${\left(T_f^m\right)}_{k,l}={\displaystyle \frac{1}{2\pi }}{\gamma }_{m,k}{\gamma }_{m,l}{\int }_0^{2\pi }{\int }_0^{+\infty }{\displaystyle \frac{f(r,\theta )r^{k+l+1}{e}^{(k-l)i\theta }drd\theta }{{(1+r^2)}^{m+2}}}.$$

3. 

${\left(T_f^m\right)}_{k,l}={\mathcal{F}\left(f({\displaystyle \frac{k}{m+1}},\theta )\right)}_{k-l}+O\left({\displaystyle \frac{1}{m}}\right).$

4. 

If $f\left(r{e}^{i\theta }\right)=f\left(r\right)$, then $T_f^m$ is a diagonal matrix.

5. 

$tr\left(T_f^m\right)=(m+1){\int }_{\mathbb{CP}\left(1\right)}fd\Omega$.

Before giving the proof, we need the following lemma.

Lemma 1.

Let $f\in C^{\infty }\left(\mathbb{CP}\left(1\right)\right)$ and let $w^k=r^k{e}^{ik\theta },w^l=r^l{e}^{il\theta }$, then

$${\int }_{\mathbb{CP}\left(1\right)}f\left(w\right){\displaystyle \frac{w^k{\overline{w}}^l}{{(1+{\left|w\right|}^2)}^m}}d\Omega ={\int }_0^{\infty }{\displaystyle \frac{r^{k+l}}{{(1+r^2)}^m}}{\int }_0^{2\pi }f(r,\theta )e^{i(k-l)\theta }{\displaystyle \frac{rdrd\theta }{{(1+r^2)}^2}}$$

Proof of Lemma 1.

This results from the continuity of the Hermitian inner product (2), the compactness of the sphere, and because the volume form $\Omega$ is finite. In fact, we have $|{(T_f^m)}_{k,l}|\le {\parallel f\parallel }_2$. □

The interest of this lemma is to justify the use of Fubini’s Theorem in our computation.

Proof. 

• The scalar product (3) applied to the sections $w^k=r^k{e}^{ik\theta },w^l=r^l{e}^{il\theta }$ gives

  $$\langle w^k,w^k\rangle =2{\int }_0^{+\infty }{\displaystyle \frac{r^{2k+1}dr}{{(1+r^2)}^{m+2}}}={\displaystyle \frac{1}{{\gamma }_{m,k}^2}}$$

  and a table of integrals gives the value stated of ${\gamma }_{m,k}$.
• We also obtain the entries of the matrix $T_f^m$:

  $$\begin{array}{cc}\hfill \langle f{w}^k,w^l\rangle & ={\displaystyle \frac{2}{2\pi }}{\gamma }_{m,k}{\gamma }_{m,l}{\int }_0^{2\pi }{\int }_0^{\infty }{\displaystyle \frac{f(r,\theta )r^{k+l+1}{e}^{(k-l)i\theta }drd\theta }{{(1+r^2)}^{m+2}}}\hfill \\ \hfill & =2{\gamma }_{m,k}{\gamma }_{m,l}{\int }_0^{\infty }{\displaystyle \frac{r^{k+l+1}}{{(1+r^2)}^{m+2}}}{\mathcal{F}\left(f\right(r,·\left)\right)}_{l-k}dr\hfill \end{array}$$
• Set

  $$\varphi (r,s)=ln(1+r^2)-2slnr$$

  $${\mathcal{L}}^m\left(g\right)\left(s\right)=2{\int }_0^{+\infty }g\left(r\right)e^{-(m+1)\varphi (r,s)}{\displaystyle \frac{r}{(1+r^2)}}dr,0\le s<1.$$
  Then, the entries of $T_f^m$ can be written as

  $${\left(T_f^m\right)}_{k,l}={\displaystyle \frac{{\mathcal{L}}^m\left(r^{l-k}{\mathcal{F}\left(f\right(r,·\left)\right)}_{l-k}\right)\left(\frac{k}{m+1}\right)}{\sqrt{{\mathcal{L}}^m\left(1\right)\left(\frac{k}{m+1}\right){\mathcal{L}}^m\left(r^{2l-2k}\right)\left(\frac{k}{m+1}\right)}}}.$$
  Since the function $\varphi (r,s)=ln(1+r^2)-2slnr$ has a local minimum at $r_0=\sqrt{{\displaystyle \frac{s}{1-s}}}$ with ${\varphi }^{″}\left(r_0\right)=4{(1-s)}^2>0$, we can apply the Laplace approximation ([13], p. 161) to the integral ${\mathcal{L}}^m\left(g\right)\left(s\right)$ to obtain

  $$\begin{array}{cc}\hfill {\mathcal{L}}^m\left(g\right)\left(s\right)& =2{\int }_0^{+\infty }g\left(r\right)e^{-(m+1)\varphi (r,s)}{\displaystyle \frac{r}{(1+r^2)}}dr\hfill \\ \hfill & =2{\displaystyle \frac{r_{0}g\left(r_0\right)}{{(1+r_0^2)}^2}}{e}^{-(m+1)\varphi (r_0,s)}\sqrt{{\displaystyle \frac{2\pi }{(m+1){\varphi }^{\prime \prime }\left(r_0\right)}}}+O({\displaystyle \frac{1}{m}}).\hfill \end{array}$$
  This leads to

  $${\left(T_f^m\right)}_{k,l}={\mathcal{F}\left(f(r_0,·)\right)}_{l-k}+O({\displaystyle \frac{1}{m}}),r_0=\sqrt{{\displaystyle \frac{\frac{k}{m+1}}{1-\frac{k}{m+1}}}}.$$
• If $f(r,\theta )=f\left(r\right),$

  $${\left(T_f^m\right)}_{k,l}=2{\gamma }_{m,k}{\gamma }_{m,l}{\int }_0^{\infty }{\displaystyle \frac{f\left(r\right)r^{k+l+1}dr}{{(1+r^2)}^{m+2}}}{\delta }_k^l.$$

  where ${\delta }_k^l$ is the Kronecker symbol.
• $$\begin{array}{cc}\hfill tr\left(T_f^m\right)=& \sum _{k=0}^{k=m}{\displaystyle \frac{1}{2\pi }}{\gamma }_{m,k}^2{\int }_0^{2\pi }{\int }_0^{\infty }{\displaystyle \frac{f(r,\theta )r^{2k+1}2drd\theta }{{(1+r^2)}^{m+2}}}\hfill \\ \hfill =& \sum _{k=0}^{k=m}{\displaystyle \frac{1}{2\pi }}(m+1)\left({\displaystyle \genfrac{}{}{0pt}{}{m}{k}}\right){\int }_0^{2\pi }{\int }_0^{\infty }{\displaystyle \frac{f(r,\theta )r^{2k}2rdrd\theta }{{(1+r^2)}^{m+2}}}\hfill \\ \hfill =& (m+1){\int }_{\mathbb{CP}\left(1\right)}fd\Omega .\hfill \end{array}$$

□

Proposition 2.

For the element of $gl\left({\Gamma }_{hol}(\mathbb{CP}\left(1\right),L^m)\right)$, we take the operator norm $\parallel ·\parallel$

$$\parallel A\parallel :=\underset{s\ne 0}{sup}{\displaystyle \frac{\parallel As\parallel }{\parallel s\parallel }}.$$

We have

$$\underset{m\to \infty }{lim}\parallel T_{fg}^m-T_f^m{T}_g^m\parallel =0.$$

Proof. 

First, let us look at the case $f(r,\theta )=h\left(r\right)e^{in\theta }.$ Recall that

$${\left(T_f^m\right)}_{k,l}={\mathcal{F}\left(f(r_k^m,·)\right)}_{l-k}+O({\displaystyle \frac{1}{m}}),r_k^m=\alpha ({\displaystyle \frac{k}{m+1}}),\alpha \left(t\right)=\sqrt{{\displaystyle \frac{t}{1-t}}}.$$

(4)

$$\begin{array}{cc}\hfill {\mathcal{F}\left(f(r_k^m,·)\right)}_{j-k}=& {\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }h\left(r_k^m\right)e^{(n+k-j)\theta }d\theta ,\hfill \\ \hfill =& h\left(r_k^m\right){\delta }_j^{n+k}.(\mathrm{Kronocker}\mathrm{symbol})\hfill \end{array}$$

Therefore,

$$\begin{array}{c}{\displaystyle {(T_f^m{T}_g^m)}_{k,l}-{(T_{fg}^m)}_{k,l}=\sum _{j=0}^{j=m}{(T_f^m)}_{k,j}{(T_g^m)}_{j,l}-{(T_{fg}^m)}_{k,l},}\hfill \\ \\ ={(T_f^m)}_{k,n+k}{(T_g^m)}_{n+k,l}-{(T_{fg}^m)}_{k,l},\hfill \\ \\ {\displaystyle ={\mathcal{F}\left(f(r_k^m,·)\right)}_n{\mathcal{F}\left(g(r_{k+n}^m,·)\right)}_{l-n-k}-\mathcal{F}{\left((hg)(r_k^m,·))\right)}_{l-k-n}+O(\frac{1}{m}),}\hfill \\ \\ {\displaystyle =h(r_k^m)\frac{1}{2\pi }{\int }_0^{2\pi }[g(r_{k+n}^m,\theta )-g(r_k^m,\theta )]e^{i(n+k-l)\theta }d\theta +O(\frac{1}{m}),}\hfill \\ \\ {\displaystyle =h(r_k^m)\frac{1}{2\pi }{\int }_0^{2\pi }\left[g(\alpha (\frac{k+n}{m+1}),\theta )-g(\alpha (\frac{k}{m+1}),\theta )\right]e^{i(n+k-l)\theta }d\theta +O(\frac{1}{m}),}\hfill \\ \\ {\displaystyle =h(r_k^m)\frac{1}{2\pi }{\int }_0^{2\pi }\left[g(\alpha (\frac{k}{m+1}+\frac{n}{m+1}),\theta )-g(\alpha (\frac{k}{m+1}),\theta )\right]e^{i(n+k-l)\theta }d\theta +O(\frac{1}{m}),}\hfill \\ \\ {\displaystyle =h(r_k^m)\frac{1}{2\pi }{\int }_0^{2\pi }\left[\frac{n}{m+1}{\alpha }^{\prime }(\frac{k}{m+1})\frac{\partial g}{\partial r}(\alpha (\frac{k}{m+1}),\theta )\right]e^{i(n+k-l)\theta }d\theta +O(\frac{1}{m}),}\hfill \\ \\ {\displaystyle =\frac{1}{m+1}{\alpha }^{\prime }(\frac{k}{m+1})\frac{1}{2\pi }{\int }_0^{2\pi }nh(r_k^m)e^{in\theta }\left[\frac{\partial g}{\partial r}(\alpha (\frac{k}{m+1}),\theta )\right]e^{i(k-l)\theta })d\theta +O(\frac{1}{m}),}\hfill \\ \\ {\displaystyle =\frac{1}{i(m+1)}{\alpha }^{\prime }(\frac{k}{m+1})\frac{1}{2\pi }{\int }_0^{2\pi }\frac{\partial f}{\partial \theta }(r_k^m,\theta )\left[\frac{\partial g}{\partial r}(\alpha (\frac{k}{m+1}),\theta )\right]e^{i(k-l)\theta }d\theta +O(\frac{1}{m}),}\hfill \\ \\ {\displaystyle =\frac{1}{i(m+1)}{\alpha }^{\prime }(\frac{k}{m+1})\frac{1}{2\pi }{\int }_0^{2\pi }\left[\frac{\partial f}{\partial \theta }\frac{\partial g}{\partial r}\right](r_k^m,\theta )e^{i(k-l)\theta }d\theta +O(\frac{1}{m}),}\hfill \\ \\ {\displaystyle =\frac{1}{i(m+1)}{\alpha }^{\prime }(\frac{k}{m+1})T_{\frac{\partial f}{\partial \theta }\frac{\partial g}{\partial r}}^m+O(\frac{1}{m}).}\hfill \end{array}$$

Now, let $f\in C^{\infty }\left(\mathbb{CP}\left(1\right)\right)$. Then, the Fourier series ${\sum }_n{f}_n\left(r\right)e^{in\theta }$ of f converges uniformly to f in the Hilbert space $L^2(\mathbb{CP}\left(1\right),d\Omega )$. So $f={\sum }_n{f}_n\left(r\right)e^{in\theta }$. Hence,

$$\begin{array}{cc}\hfill {(T_f^m{T}_g^m)}_{k,l}-{(T_{fg}^m)}_{k,l}=& \underset{n\to \infty }{lim}{\Sigma }_{-n}^n\left[{(T_{f_n}^m{T}_g^m)}_{k,l}-{(T_{f_{n}g}^m)}_{k,l}\right]+O({\displaystyle \frac{1}{m}}),\hfill \\ \hfill =& \underset{n\to \infty }{lim}{\Sigma }_{-n}^n{\left[{\alpha }^{\prime }({\displaystyle \frac{k}{m+1}}){\displaystyle \frac{-i}{m+1}}{T}_{{\displaystyle \frac{\partial f_n}{\partial \theta }}{\displaystyle \frac{\partial g}{\partial r}}}^m\right]}_{k,l}+O({\displaystyle \frac{1}{m}}),\hfill \\ \hfill =& {\alpha }^{\prime }({\displaystyle \frac{k}{m+1}}){\displaystyle \frac{-i}{m+1}}{\left[T_{{lim}_{n\to \infty }{\Sigma }_{-n}^n{\displaystyle \frac{\partial f_n}{\partial \theta }}{\displaystyle \frac{\partial g}{\partial r}}}^m\right]}_{k,l}+O({\displaystyle \frac{1}{m}})\hfill \\ \hfill =& {\alpha }^{\prime }({\displaystyle \frac{k}{m+1}}){\displaystyle \frac{-i}{m+1}}{\left[T_{{\displaystyle \frac{\partial f}{\partial \theta }}{\displaystyle \frac{\partial g}{\partial r}}}^m\right]}_{k,l}+O({\displaystyle \frac{1}{m}}).\hfill \end{array}$$

So, we have proved

$${\left[T_f^m{T}_g^m-T_{fg}^m\right]}_{k,l}={\alpha }^{\prime }({\displaystyle \frac{k}{m+1}}){\displaystyle \frac{-i}{m+1}}{T}_{{\displaystyle \frac{\partial f}{\partial \theta }}{\displaystyle \frac{\partial g}{\partial r}}}^m+O({\displaystyle \frac{1}{m}}),$$

(5)

from which we deduce

$$\underset{m\to \infty }{lim}\parallel T_f^m{T}_g^m-T_{fg}^m\parallel =0.$$

□

Theorem 1.

If f and g are two $C^{\infty }$ functions on $\mathbb{CP}\left(1\right)$, then

$$\parallel i(m+1)[T_f^m,T_g^m]-T_{\{f,g\}}^m\parallel =O\left(m^{-1}\right).$$

Had we denoted $T_f^m$ by $T_f^{m+1},$ we would have $\parallel im[T_f^m,T_g^m]-T_{\{f,g\}}^m\parallel =O\left(m^{-1}\right)$.

Proof. 

We have seen in relation (5) that

$${\left[T_f^m{T}_g^m-T_{fg}^m\right]}_{k,l}={\alpha }^{\prime }({\displaystyle \frac{k}{m+1}}){\displaystyle \frac{-i}{m+1}}{T}_{{\displaystyle \frac{\partial f}{\partial \theta }}{\displaystyle \frac{\partial g}{\partial r}}}^m+O({\displaystyle \frac{1}{m}}).$$

Recall that the Poisson bracket in polar coordinates is given by:

$$\{f,g\}={\displaystyle \frac{{(1+r^2)}^2}{2r}}\left[{\displaystyle \frac{\partial f}{\partial r}}{\displaystyle \frac{\partial g}{\partial \theta }}-{\displaystyle \frac{\partial g}{\partial r}}{\displaystyle \frac{\partial f}{\partial \theta }}\right].$$

Recall also from relation (4) that since $r=\alpha \left(t\right)=\sqrt{{\displaystyle \frac{t}{1-t}}}$, then

$${\alpha }^{\prime }\left(t\right)={\displaystyle \frac{{(1+r^2)}^2}{2r}}.$$

As a consequence of the above considerations, we have

$$\begin{array}{cc}\hfill {\left[T_f^m{T}_g^m-T_g^m{T}_f^m\right]}_{k,l}=& {\left[T_f^m{T}_g^m-T_{fg}^m\right]}_{k,l}-{\left[T_g^m{T}_f^m-T_{fg}^m\right]}_{k,l}\hfill \\ \hfill =& \left[{\alpha }^{\prime }({\displaystyle \frac{k}{m+1}}){\displaystyle \frac{-i}{m+1}}{T}_{{\displaystyle \frac{\partial f}{\partial \theta }}{\displaystyle \frac{\partial g}{\partial r}}}^m\right]-\left[{\alpha }^{\prime }({\displaystyle \frac{k}{m+1}}){\displaystyle \frac{-i}{m+1}}{T}_{{\displaystyle \frac{\partial g}{\partial \theta }}{\displaystyle \frac{\partial f}{\partial r}}}^m\right]+O({\displaystyle \frac{1}{m}})\hfill \\ \hfill =& {\displaystyle \frac{-i}{m+1}}{\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }{\alpha }^{\prime }(r_k^m)\left[{\displaystyle \frac{\partial f}{\partial r}}{\displaystyle \frac{\partial g}{\partial \theta }}-{\displaystyle \frac{\partial g}{\partial r}}{\displaystyle \frac{\partial f}{\partial \theta }}\right](r_k^m,\theta )e^{i(k-l)\theta }d\theta +O({\displaystyle \frac{1}{m}})\hfill \\ \hfill =& {\displaystyle \frac{-i}{m+1}}{\left[T_{\{f,g\}}^m\right]}_{k,l}+O({\displaystyle \frac{1}{m}}).\hfill \end{array}$$

We conclude then that

$$i(m+1){\left[T_f^m{T}_g^m-T_g^m{T}_f^m\right]}_{k,l}-{\left[T_{\{f,g\}}^m\right]}_{k,l}=O({\displaystyle \frac{1}{m}})$$

which leads to

$$\parallel i(m+1)\left[T_f^m,T_g^m\right]-T_{\{f,g\}}^m\parallel =O({\displaystyle \frac{1}{m}}).$$

□

Remark 1.

Let $\gamma \left(t\right)=e^{-it{T}_f^m}{T}_g^m{e}^{it{T}_f^m}-T_{F_t^{*}\left(g\right)}^m$ be the curve in the space of Hermitian matrices ${\mathcal{H}}_m$. We have

$${\gamma }^{\prime }\left(0\right)=im[T_f^m,T_g^m]-T_{\{f,g\}}^m.$$

As a consequence of proposition (2), we give here a new proof of the following generalization of Szegö’s Theorem due to V. Guillemin [14] in the case of the sphere.

Theorem 2

(Szegö). Let f be a smooth real-valued function on $\mathbb{CP}\left(1\right)$, let $T_f^m$ be the Toeplitz quantization of f, and let ${\mu }_m$ be the spectral measure of $T_f^m$. Then, as m tends to infinity, ${\displaystyle \frac{{\mu }_m}{m+1}}$ converges weakly to the measure, μ, defined by

$${\displaystyle \mu \left(\varphi \right)={\int }_{\mathbb{CP}\left(1\right)}\varphi \left(f\right)d\Omega ,\forall \varphi \in \mathcal{C}\left(\mathbb{R}\right).}$$

In other words, if ${\left({\lambda }_k\left(T_f^m\right)\right)}_k$ are the eigenvalues of the matrix $T_f^m$, then

$${\displaystyle {\int }_{\mathbb{CP}\left(1\right)}\varphi \left(f\right)d\Omega =\underset{m\mapsto \infty }{lim}\frac{{\sum }_{k=0}^{k=m}\varphi \left({\lambda }_k\left(T_f^m\right)\right)}{m+1}.}$$

For the proof, we need to review material about asymptotically equivalent matrices that can be found in ([15], p. 62 and [16], p. 17).

If A is an $m\times m$ diagonalizable matrix, then we will indicate by ${\lambda }_k\left(A\right),1\le k\le m$ all the eigenvalues of A counted with their multiplicities. The following matrix norms will be used:

Definition 2.

The Frobenius norm ${\parallel A\parallel }_F$ and the spectral norm ${\parallel A\parallel }_2$ of an $m\times m$ matrix $A={\left(a_{j,k}\right)}_{j,k=1}^m$ are defined as

$${\parallel A\parallel }_F=\sqrt{tr\left(A{A}^{*}\right)}={\left(\sum _{k=1}^{k=m}\sum _{j=1}^{j=m}{\left|a_{j,k}\right|}^2\right)}^{{\displaystyle \frac{1}{2}}}$$

$${\parallel A\parallel }_2=\underset{x\ne 0}{max}{\left({\displaystyle \frac{x^{*}{A}^{*}Ax}{x^{*}x}}\right)}^{{\displaystyle \frac{1}{2}}}={\left(\underset{1\le k\le m}{max}{\lambda }_k\left(A{A}^{*}\right)\right)}^{{\displaystyle \frac{1}{2}}}$$

where $tr$ denotes trace, * denotes conjugate transpose, and $x^{*}$ is a column vector.

The next definition is due to Gray ([16], p. 17).

Definition 3.

Two sequences of Hermitian matrices $\left\{A_m\right\}$ and $\left\{B_m\right\}$ are asymptotically equivalent, if

$$\exists K\ge 0; \parallel A_m{\parallel }_2, {\parallel B_m\parallel }_2\le K, \forall m\in \mathbb{N}$$

and

$$\underset{m\to \infty }{lim}{\displaystyle \frac{\parallel A_m-B_m{\parallel }_F}{\sqrt{m}}}=0.$$

The following proposition illustrates the behavior of the eigenvalues of asymptotically equivalent matrices.

Proposition 3

([16]). If two sequences of Hermitian matrices $\left\{A_m\right\}$ and $\left\{B_m\right\}$ with respective eigenvalues ${\left({\lambda }^k\left(A_m\right)\right)}_k$ and ${\left({\lambda }^k\left(B_m\right)\right)}_k$ are asymptotically equivalent, then the sequences of their eigenvalues are equally distributed: there exists $K>0$ such that $|{\lambda }^k\left(A_m\right)|<K$ and $\left|\right({\lambda }^k\left(B_m\right)\left)\right|<K$ and for every continuous function ϕ on $[-K.K]$, we have

$$\underset{m\to \infty }{lim}{\displaystyle \frac{{\sum }_{k=0}^m\left[\varphi \left({\lambda }^k\left(A_m\right)\right)-\varphi \left({\lambda }^k\left(B_m\right)\right)\right]}{m+1}}=0.$$

(6)

Theorem 3.

Let f be a $C^{\infty }$ function and let $p\in \mathbb{N}$ be a natural number. The matrices ${(T_{f^p}^m)}_m$ and ${({(T_f^m)}^p)}_m$ are asymptotically equivalent.

Proof. 

We know from Proposition 1 that

$$\underset{m\to \infty }{lim}\parallel T_{f^2}^m-{\left(T_f^m\right)}^2\parallel =0.$$

But for Hermitian matrices, $\parallel T_{f^2}^m-{\left(T_f^m\right)}^2\parallel ={\parallel T_{f^2}^m-{\left(T_f^m\right)}^2\parallel }_2$. So

$$\underset{m\to \infty }{lim}{\parallel T_{f^2}^m-{\left(T_f^m\right)}^2\parallel }_2=0.$$

It remains to prove that $\parallel {\lambda }_k(T_{f^2}^m){\parallel \le \parallel f\parallel }_{\infty }^2$ and $\parallel {\lambda }_k(T_f^m){\parallel \le \parallel f\parallel }_{\infty }$. This results from the following lemma. □

Lemma 2.

Let $g\in C^{\infty }\left(\mathbb{CP}\left(1\right)\right)$ and let μ be an eigenvalue of $T_g^m$. Then

$$\left|\mu \right|\le {\parallel g\parallel }_{\infty }.$$

Proof. 

Let v be a unit eigenvector associated to $\mu$. We have, using the scalar product (3) and the metric (2),

$$\mu =\langle T_g^{m}v,v\rangle ={\int }_{\mathbb{CP}\left(1\right)}g{h}^m(v,v)d\Omega \le {\parallel g\parallel }_{\infty }\langle v,v\rangle ={\parallel g\parallel }_{\infty }.$$

We deduce that the matrices ${({(T_f^m)}^2)}_m$ and ${(T_{f^2}^m)}_m$ are asymptotically equivalent.

By induction, we prove that the matrices ${(T_{f^p}^m)}_m$ and ${({\left[T_f^m\right]}^p)}_m$ are also asymptotically equivalent. Indeed, we have

$$\begin{array}{cc}\hfill \parallel T_{f^p}^m-{[T_f^m]}^p\parallel \le & \parallel T_{f^p}^m-T_{f^{p-1}}^m{T}_f^m\parallel +\parallel T_{f^{p-1}}^m{T}_f^m-{[T_f^m]}^p\parallel ,\hfill \\ \hfill \le & \parallel T_{f{f}^{p-1}}^m-T_{f^{p-1}}^m{T}_f^m\parallel +\parallel T_f^m\parallel \parallel T_{f^{p-1}}^m-{[T_f^m]}^{p-1}\parallel ,\hfill \\ \hfill \le & \parallel T_{f{f}^{p-1}}^m-T_{f^{p-1}}^m{T}_f^m\parallel +{\parallel f\parallel }_{\infty }\parallel [T_{f^{p-1}}^m-{[T_f^m]}^{p-1}\parallel .\hfill \end{array}$$

The first term goes to zero when m goes to infinity and is the result of applying Theorem 2 to the functions f and $g=f^{p-1}$, whereas the second term goes to zero when m goes infinity by induction on $p\in \mathbb{N}$.

Now, since the matrices ${(T_{f^p}^m)}_m$ and ${({[T_f^m]}^p)}_m$ are asymptotically equivalent, their eigenvalues are equally distributed. Therefore, taking $\varphi \left(x\right)=x$ in Proposition 3, we obtain

$$\begin{array}{cc}\hfill \underset{m\to \infty }{lim}{\displaystyle \frac{{\sum }_{k=0}^{k=m}{\lambda }_k(T_{f^p}^m)}{m+1}}=& \underset{m\to \infty }{lim}{\displaystyle \frac{{\sum }_{k=0}^{k=m}{\lambda }_k({(T_f^m)}^p)}{m+1}},\hfill \\ \hfill ={\int }_{\mathbb{CP}(1)}{f}^{p}d\Omega =& \underset{m\to \infty }{lim}{\displaystyle \frac{{\sum }_{k=0}^{k=m}{[{\lambda }_k(T_f^m)]}^p}{m+1}}.\hfill \end{array}$$

From the last equality, we see that if $P(z)$ is a polynomial, then

$${\int }_{\mathbb{CP}(1)}P(f)d\Omega =\underset{m\mapsto \infty }{lim}{\displaystyle \frac{{\sum }_{k=0}^{k=m}P({\lambda }_k(T_f^m))}{m+1}}.$$

Also, since every continuous function on $\mathbb{CP}\left(1\right)$ can be approximated uniformly by polynomials (Stone–Weierstrass Theorem), we have:

$${\int }_{\mathbb{CP}\left(1\right)}\varphi \left(f\right)d\Omega =\underset{m\mapsto \infty }{lim}{\displaystyle \frac{{\sum }_{k=0}^{k=m}\varphi ({\lambda }_k(T_f^m))}{m+1}},\varphi \in \mathcal{C}\left(\mathbb{R}\right).$$

□

3. Convergence of Eigenvalues of ${\mathit{T}}_{\mathit{f}}^{\mathit{m}}$

Before looking at the asymptotic behavior of the eigenvalues of the Hermitian matrix $T_f^m$, let us review quickly the notion of decreasing rearrangement of a function. This is a deep generalization of the simple act of arranging a finite list of numbers in decreasing order. A function is a list of continuously many numbers, and it may be useful, in certain applications, to rearrange those in a different order, for instance, because comparisons between complicated functions become feasible.

Let $f:(X,\mu )\to \mathbb{R}$ be a measurable function. Let $d_f\left(t\right)=\mu \left(\{\omega :f\left(\omega \right)\ge t\}\right)$ be the distribution function of f.

The decreasing rearrangement $f^{*}$ of the function f is

$$f^{*}\left(x\right)=inf\{t:d_f\left(t\right)<x\}=sup\{t:d_f\left(t\right)>x\},0\le x\le 1$$

Because of the importance of $f^{*}$ in the sequel, we will follow [17,18] and describe how to construct the decreasing rearrangement of a step function.

Example 1.

Let $f\left(x\right)={\Sigma }_j{u}_j{\chi }_{E_j}\left(x\right),$ where $\left\{E_j\right\}$ is a partition of $[0,1]$. We order the values of f in a decreasing order: $t_1>t_2>\cdots >t_p,$ and to each value $t_j$, we associate the set $F_j=f^{-1}\left(t_j\right)$. Let

$$a_0=0,a_1=\left|F_1\right|,a_2=\left|F_1\right|+\left|F_2\right|,\cdots a_p=\sum _1^p\left|F_k\right|=1.$$

With $t_1=maxf{\left(x\right)}_{x\in [0,1]},t_p=minf{\left(x\right)}_{x\in [0,1]},$ we obtain

$$f^{*}\left(s\right)=\left\{\begin{array}{ccccc}{t}_1& & 0\le & s& <a_1\\ t_2& & a_1\le & s& <a_2\\ ⋮& & ⋮& \cdots & <⋮\\ t_p& & a_{p-1}\le & s& \le 1\end{array}\right.$$

(7)

Remark 2.

The main properties of the decreasing rearrangement that we need later are: for every continuous ϕ defined on the range of the function f

1. 

${\displaystyle {\int }_{\mathbb{CP}\left(1\right)}\varphi \left(f\right)d\Omega ={\int }_0^1\varphi \left(f^{*}\right)\left(t\right)dt.}$

2. 

${\displaystyle {\int }_{\mathbb{CP}\left(1\right)}\varphi \left(f\right)d\Omega ={\int }_{\mathbb{CP}\left(1\right)}\varphi \left(g\right)d\Omega \Rightarrow f^{*}=g^{*}.}$

3. 

${sup\left|f\left(x\right)\right|}_{x\in X}=sup{\left|f^{*}\left(t\right)\right|}_{t\in [0,1]}$.

For more information on decreasing rearrangement of a function, one can also see [17].

One of our fundamental results is the striking fact that the decreasing rearrangement of the function f is determined by the decreasing rearrangement of the eigenvalues of the matrix $T_f^m$ as it is shown by the following theorem.

Theorem 4.

Let $T_f^m$ be the Toeplitz quantization of $f\in C^{\infty }\left(\mathbb{CP}\left(1\right)\right)$ and let ${\lambda }^m=\left({\lambda }_0^m,{\lambda }_1^m,\cdots ,{\lambda }_m^m\right)$ be the eigenvalues of $T_f^m$ arranged in non-increasing order. Let ${\Lambda }^m\left(r\right)$ be the real step function defined on the interval $[0,1]$ by

$${\Lambda }^m\left(\left[{\displaystyle \frac{k}{m+1}},{\displaystyle \frac{k+1}{m+1}}\right[\right)={\lambda }_k^m,0\le k\le m$$

Then, the sequence of decreasing rearrangement of the functions ${\Lambda }^m\left(r\right)$ converges point-wise almost everywhere to the decreasing rearrangement $f^{*}$ of the function f.

Proof. 

Szegö’s Theorem (2) says for every continuous function $\varphi$, defined on the image of the function f, we have

$${\displaystyle \underset{m\mapsto +\infty }{lim}\sum _{k=0}^{k=m}\frac{1}{m+1}\varphi \left({\lambda }_k^m\right)={\int }_{\mathbb{CP}\left(1\right)}\varphi \left(f\right)d\Omega .}$$

(8)

However, since f and $f^{*}$ are equi-measurable, we have

$${\int }_{\mathbb{CP}\left(1\right)}\varphi \left(f\right)d\Omega ={\int }_0^1\varphi \circ f^{*}\left(t\right)dt.$$

Relation (8) becomes

$$\begin{array}{cc}\hfill {\displaystyle \underset{m\mapsto +\infty }{lim}\sum _{k=0}^{k=m}\frac{1}{m+1}\varphi \left({\lambda }_k^m\right)=}& {\displaystyle {\int }_0^1\varphi \circ f^{*}\left(t\right)dt,}\\ \hfill {\displaystyle \underset{m\mapsto +\infty }{lim}{\int }_0^1\varphi \left({\Lambda }^m\right)\left(t\right)dt=}& {\displaystyle {\int }_0^1\varphi \left(f^{*}\right)\left(t\right)dt.}\end{array}$$

(9)

Relation (9) is equivalent to: the sequence of state functions ${\left({\Lambda }^m\right)}_m$ converges in distribution to the real function $f^{*}.$ Ref. [19] (p. 84) is a good reference for more details on convergence in distribution).

A well-known theorem by Skorokhod [17] states that sequence of the decreasing rearrangement of ${\left({\Lambda }^m\right)}_m$ converges point-wise to the decreasing rearrangement of $\left(f^{*}\right)$. However, since $f^{*}$ and ${\Lambda }^m$ are decreasing functions, we have ${\left(f^{*}\right)}^{*}=f^{*}$, and ${\left({\Lambda }^m\right)}^{*}={\Lambda }^m$.

Consequently, the sequence ${\left({\Lambda }^m\right)}_m$ converges point-wise almost everywhere to $f^{*}$. □

Example 2.

Let $J\left(w\right)=J\left(r{e}^{i\theta }\right)={\displaystyle \frac{1}{1+r^2}}$. Because J depends only on r, $T_J^m$ is a diagonal matrix and its spectrum is

$$\begin{array}{ccc}\hfill {\left(T_J^m\right)}_{k,k}={\lambda }_k^m& =& {\left({\gamma }_{m,k}\right)}^{2}2{\int }_0^{+\infty }{\displaystyle \frac{r^{2k+1}dr}{{(1+r^2)}^{m+3}}},\hfill \\ & =& {\displaystyle \frac{2{\left({\gamma }_{m,k}\right)}^2}{2{\left({\gamma }_{m+1,k}\right)}^2}},\hfill \\ & =& {\displaystyle \frac{(m+1)\left(\genfrac{}{}{0pt}{}{m}{k}\right)}{(m+2)\left(\genfrac{}{}{0pt}{}{m+1}{k}\right)}},\hfill \\ & =& {\displaystyle \frac{m+1-k}{m+2}},\hfill \end{array}$$

(10)

and if we set $s=k\setminus m,$ in (10), then

$${\Lambda }^m\left(s\right)={\lambda }_k^m={\displaystyle \frac{1+{\displaystyle \frac{1}{m}}-s}{1+{\displaystyle \frac{2}{m}}}}$$

from which we obtain ${lim}_{m\mapsto +\infty }{\Lambda }^m\left(s\right)=1-s$ or $J^{*}\left(s\right)=1-s$.

Remark 3.

Theorem (4) says a lot more. In our paper [20], we have shown that one can think of a function $f(r,\theta )$ on the sphere as an infinite matrix where the entries are indexed along the ${\theta }^{th}$ diagonal by r and the decreasing rearrangement of f is therefore its spectrum.

Theorem 5.

For every $f\in C^{\infty }\left(\mathbb{CP}\left(1\right)\right)$, we have

$$\underset{m\to \infty }{lim}\parallel T_f^m\parallel ={\parallel f\parallel }_{\infty }.$$

Proof. 

With the notation of Theorem (4),we have

$$\parallel T_f^m\parallel =max\{\left|{\lambda }_k^m\right|,0\le k\le m\}=sup\left|{\Lambda }_m\right|.$$

Therefore,

$$\underset{m\to \infty }{lim}\parallel T_f^m\parallel =\underset{m\to \infty }{lim}sup\left|{\Lambda }_m\right|={\parallel f^{*}\parallel }_{\infty }={\parallel f\parallel }_{\infty }.$$

□

4. Convergence of the Spectrum of the Toeplitz Quantization Kähler Manifold

Let X be an n-dimensional non-singular complex projective variety embedded in $\mathbb{CP}\left(n\right)$. Let L be the canonical line bundle on X. There is a unique connection on L with the property that if s is a local non-vanishing holomorphic section, $Ds=\overline{\partial }s=ln{\left|s\right|}^2\otimes s$.

If $\omega$ is the curvature form of this connection, then $(X,\omega )$ is a Kähler manifold. Let $\nu =\omega \wedge \omega \wedge \cdots \wedge \omega$ (n wedges) be the volume form. The space of smooth sections of ${⨂}^m{L}^{*}$ has an intrinsic pre-Hilbert structure. Let ${\pi }_m$ be the orthogonal projection of this space onto, ${\mathcal{H}}^m$, the subspace of holomorphic sections.

Let also recall that for large m, the dimension of the space of holomorphic sections of ${⨂}^m{L}^{*}$ is

$${\displaystyle \frac{\mathrm{vol}\left(X\right)}{\mathrm{vol}\left(\mathbb{CP}\right(n\left)\right)}}{\displaystyle \frac{m^n}{n!}}+O\left(m^{n-1}\right).$$

Given a real valued function f on X, let $M_f$ be the operator “multiplication by f” on the space of smooth section of ${⨂}^m{L}^{*}$ and let $T_f^m={\pi }_m{M}_f{\pi }_m$ be the Toeplitz quantization of f.

Theorem 6.

Let ${\lambda }^m=\left({\lambda }_0^m,{\lambda }_1^m,\cdots ,{\lambda }_m^m\right)$ be the eigenvalues of $T_f^m$ arranged in non-increasing order.

Let ${\Lambda }^m\left(t\right)$ be the real step function defined on the interval $[0,1]$ by

$${\Lambda }^m\left(t\right)={\lambda }_k^m,\mathrm{if}{\displaystyle \frac{k}{dim\left({\mathcal{H}}^m\right)}}\le t<{\displaystyle \frac{k+1}{dim\left({\mathcal{H}}^m\right)}}.$$

Then, the sequence of decreasing rearrangement of the functions ${\Lambda }^m\left(r\right)$ converges point-wise almost everywhere to the decreasing rearrangement $f^{*}$ of the function f.

Proof. 

The proof is very much identical to the case of the sphere V. Guillemin proved in [14] the following theorem. Let ${\mu }_m$ be the spectral measure of ${\pi }_m{M}_f{\pi }_m$. As m tends to infinity, ${\displaystyle \frac{{\mu }_m}{m^n}}$ converges to the measure $\mu$ defined by

$$\mu \left(\varphi \right)={\displaystyle \frac{1}{n!{\nu }_n}}{\int }_X\varphi \left(f\left(x\right)\right)d\nu ,\forall \varphi \in C\left(\mathbb{R}\right).$$

(11)

where ${\nu }_n$ is the volume of $\mathbb{CP}\left(n\right)$. Relation (11) can be written

$$\underset{m\to \infty }{lim}{\displaystyle \frac{{\sum }_{k=0}^{k=dim\left({\mathcal{H}}^m\right)}\varphi \left({\lambda }_k^m\right)}{m^n}}={\displaystyle \frac{1}{n!{\nu }_n}}{\int }_X\varphi \left(f\left(x\right)\right)d\nu ,\forall \varphi \in C\left(\mathbb{R}\right).$$

Or equivalently,

$$\underset{m\to \infty }{lim}{\displaystyle \frac{{\sum }_{k=0}^{k=dim\left({\mathcal{H}}^m\right)}\varphi \left({\lambda }_k^m\right)}{dim\left({\mathcal{H}}^m\right)}}={\displaystyle \frac{1}{\mathrm{vol}\left(X\right)}}{\int }_X\varphi \left(f\left(x\right)\right)d\nu ,\forall \varphi \in C\left(\mathbb{R}\right).$$

(12)

Or

$$\underset{m\to \infty }{lim}{\int }_0^1\varphi \left({\Lambda }^m\right)dt={\displaystyle \frac{1}{\mathrm{vol}\left(X\right)}}{\int }_X\varphi \left(f\left(x\right)\right)d\nu ,\forall \varphi \in C\left(\mathbb{R}\right).$$

(13)

Let $f^{*}$ be the decreasing rearrangement of the function f on the measure space $(X,{\nu }^{\prime })$, where ${\nu }^{\prime }={\displaystyle \frac{\nu }{\mathrm{vol}\left(X\right)}}$ the scaled volume. Then, relation (13) can be written again as $\forall \varphi \in C\left(\mathbb{R}\right)$,

$$\begin{array}{cc}\hfill \underset{m\to \infty }{lim}{\int }_0^1\varphi \left({\Lambda }^m\right)dt=& {\displaystyle \frac{1}{\mathrm{vol}\left(X\right)}}{\int }_X\varphi \left(f\left(x\right)\right)d\nu ,\\ \hfill =& {\int }_0^1\varphi \left(f^{*}\right)dt\end{array}$$

But this means that the sequence of functions ${\left({\Lambda }^m\right)}_m$ converges in distribution to $f^{*}$. Again, this implies by using Skorokhod’s theorem ([17]), that the sequence ${\left({\Lambda }^m\right)}_m$ converges a.e to $f^{*}.$ □

5. Conclusions

On the sphere $\left(\mathbb{CP}\right(1),\Omega )$, we have seen how $C^{\infty }$ functions behave like Hermitian matrices.

In [1,2,21,22], authors have shown similarities between Toda Lattice equations and the dispersionless Toda equations.

A Toda Lattice equation is the matrix equation

$$\dot{L}\left(t\right)=\left[B\left(t\right),L\left(t\right)\right]=B\left(t\right)L\left(t\right)=L\left(t\right)B\left(t\right),$$

(14)

where

$$L=\left(\begin{array}{ccccc}{b}_1& a_1& 0& \cdots & 0\\ a_1& b_2& a_2& \cdots & 0\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ 0& \cdots & \ddots & b_{n-1}& a_{n-1}\\ 0& \cdots & \cdots & a_{n-1}& 0\end{array}\right)$$

(15)

and

$$B=\left(\begin{array}{ccccc}0& a_1& 0& \cdots & 0\\ -a_1& 0& a_2& \cdots & 0\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ 0& \cdots & \ddots & 0& a_{n-1}\\ 0& \cdots & \cdots & -a_{n-1}& 0\end{array}\right).$$

(16)

The dispersionless Toda equation is

$${\displaystyle \frac{\partial v}{\partial t}}=v{\displaystyle \frac{\partial u}{\partial z}},{\displaystyle \frac{\partial u}{\partial t}}=2{\displaystyle \frac{\partial v^2}{\partial z}}.$$

(17)

if $L=u\left(z\right)+2v\left(z\right)cos\left(\theta \right)$ and if $B=-2v\left(z\right)sin\left(\theta \right)$, then the Equation (17) becomes

$${\displaystyle \frac{dL}{dt}}=\{L,B\}.$$

(18)

Now, let $L=u\left(z\right)+2v\left(z\right)cos\left(\theta \right)$ and $B=-2v\left(z\right)sin\left(\theta \right)$ and let us form the sequence of Toda latice equtions

$${\displaystyle \frac{d\left(T_L^m\right)}{dt}}=[T_L^m,T^{m}B].$$

It seems reasonable, as in [23], from all that we have seen, to think that Toeplitz quantization can be used to solve the dispersionless Toda equation from the solution of the Toda lattice equation.

Mathematics and physics have been affected significantly by Toeplitz quantization, particularly geometric quantization, which is an important framework in quantum mechanics and mathematical physics. Quantum systems are constructed based on the geometry of phase space, a method closely related to geometric quantization. An implementation of geometric quantization using Toeplitz quantization, particularly for Kähler manifolds, offers a concrete application [22]. Toeplitz operators on Hilbert spaces are used in the framework because they are bounded linear operators on Hilbert spaces. It is possible to study their spectral properties and their role in quantum mechanics by quantizing these operators, as they have well-defined properties that make them suitable for quantization [24]. Researchers are investigating the mathematical foundations and applications of Toeplitz quantization, as well as its connections to other fields of physics and mathematics. There is a possibility that this research will lead to new insights and advancements in both fields. Furthermore, Toeplitz quantization contributes to various areas of mathematical physics and operator theory through its connection between classical and quantum mechanics, geometric quantization, and its applications in geometric quantization. We continue to learn more about quantum systems and their underlying mathematical structures through their ongoing research in [6,21,25,26,27,28,29].