Analytic Berezin–Toeplitz operators

Abstract

We introduce new tools for analytic microlocal analysis on Kähler manifolds. As an application, we prove that the space of Berezin–Toeplitz operators with analytic contravariant symbol is an algebra. We also give a short proof of the Bergman kernel asymptotics up to an exponentially small error.

Appendix

In the first part, we discuss the asymptotic expansion of analytic symbols. We work in an abstract setting where the symbols are not functions but belong to a normed space, because it makes the discussion simpler. One goal is to compare some remainder estimates (34) coming from [5] with the partial sums (35) introduced in [16]. Even if we haven’t found such a discussion in the literature, we do not claim that these results are original.

The second part is a digression on the method we use to prove Theorems 3.1 and  5.1. We propose alternative arguments and some generalisation.

1.1 Analytic asymptotic expansion

Let $(E, | \cdot |)$ be a normed space. Consider a sequence (u(k)) of E having an asymptotic expansion of the form

$$\begin{aligned} u(k) = \sum _{\ell =0 }^{N-1} a_\ell k^{-\ell } + \mathcal {O}( k^{-N}) , \qquad \forall N \end{aligned}$$

(33)

with coefficients $a_\ell \in E$. Two important facts are that (u(k)) is determined modulo $\mathcal {O}( k^{-\infty })$ by the $a_\ell$’s, and for any sequence $(a_{\ell })$, there exists a sequence (u(k)) satisfying (33).

We are interested in a particular class of asymptotic expansions where the remainder in (33) has the following explicit upper bound

$$\begin{aligned} \bigl | u(k) - \sum _{\ell =0 }^{N-1} a_\ell k^{-\ell } \bigr | \leqslant C^{N+1} k^{-N} N! , \qquad \forall \; k\in \mathbb {N} ^*,\; N \in \mathbb {N} \end{aligned}$$

(34)

for a constant C independent of k and N. Unlike the expansion (33), the sequence (u(k)) is uniquely determined up to a $\mathcal {O}(e^{-\epsilon k})$ by (34). Furthermore, the coefficients $a_{\ell }$ have a particular growth. The precise result is as follows.

Proposition 5.3

• 1. If a sequence (u(k)) satisfies (34), then the coefficients $(a_\ell )$ satisfy $|a_\ell | \leqslant C^{\ell +1} \ell !$ with the same constant C.

• 2. Assume that (u(k)) satisfies (34). Then $(u'(k))$ satisfies (34) with the same coefficients $a_\ell$ and possibly a larger constant C if and only if there exists $\epsilon >0$ such that $u (k ) = u'( k) + \mathcal {O}( e^{-\epsilon k})$.

• 3. If $|a_{\ell } | \leqslant (C')^{\ell +1} \ell !$ for $C'>0$ and $\epsilon >0$ is such that $\epsilon C' <1$, then

  $$\begin{aligned} u (k) := \sum _{\ell =0 }^{\lfloor \epsilon k \rfloor } a_{\ell } k^{-\ell } \end{aligned}$$

  (35)

  satisfies (34) for some $C>0$.

Proof

1. (34) implies that $|a_N k^{-N} | \leqslant C^{N+1} k^{-N} N! + C^{N+2} k^{-N-1} (N+1)!$. Multiplying by $k^N$ and taking the limit $k \rightarrow \infty$, we get $|a_N | \leqslant C^{N+1} N!$.

2. Assume that (u(k)) and $(u'(k))$ satisfy both (34). Then

$$\begin{aligned} | u (k) - u'(k) | \leqslant 2C^{N+1} k^{-N} N! \leqslant 2 C (CN/k)^N. \end{aligned}$$

Choose $\epsilon$ such that $C \epsilon <1$ and set $N =\lfloor \epsilon k \rfloor$. Then $N \leqslant \epsilon k$ so $CN/k \leqslant C \epsilon$ so $(CN/k)^N \leqslant (C \epsilon )^N \leqslant (C\epsilon )^{\epsilon k - 1 }$ because $N \geqslant \epsilon k -1$. So $|u (k) - u'(k)| \leqslant 2 \epsilon ^{-1} \exp ( \epsilon k \ln ( C \epsilon ) )$. Hence $u (k) = u'(k) +\mathcal {O}( e^{-\epsilon ' k})$ with $\epsilon ' = - \epsilon \ln (C\epsilon )$. Conversely, we have to prove that for any $\epsilon$, there exists $C >0$ such that $e^{-\epsilon k } \leqslant C (CN/k)^N$ for any $k \in \mathbb {N} ^*$ and $N\in \mathbb {N}$. The function $x \rightarrow \ln x - \epsilon x$ is bounded above on $\mathbb {R} _{>0}$, so

$$\begin{aligned} \ln (k/N) - \epsilon (k/N) \leqslant \ln C, \qquad \forall \; k,N \in \mathbb {N} ^* \end{aligned}$$

if C is sufficiently large. Multiplying with N and taking the exponential, we get $e^{-\epsilon k} \leqslant (CN/k)^N$. We conclude easily.

3. By assumption $| a_{\ell } k^{-\ell } | \leqslant C' (C'/k)^\ell \ell ! =: b_{\ell }$. We will use that

$$\begin{aligned} \frac{b_\ell }{b_{\ell -1}} = C' \ell / k. \end{aligned}$$

Let $N' =\lfloor \epsilon k \rfloor$. Assume that $N \leqslant \ell \leqslant N'$. Then $\ell \leqslant \epsilon k$, so $b_{\ell } / b_{\ell -1} \leqslant C'\epsilon <1$, so

$$\begin{aligned} \sum _{\ell = N}^{N'} b_{\ell } \leqslant b_N ( 1 + \cdots + (C'\epsilon )^{N' - N} ) \leqslant \frac{b_N}{1 - C'\epsilon } = \frac{C'}{1 - C' \epsilon } (C'/k)^N N! \end{aligned}$$

Assume now that $N' < \ell \leqslant N$. Then $\epsilon k \leqslant \ell$, so $b_{\ell }/ b_{\ell +1 } \leqslant (C'\epsilon )^{-1} =: r$. Since $r>1$,

$$\begin{aligned} \sum _{\ell = N'+1}^{N-1} b_{\ell }\leqslant & {} \sum _{\ell = N'+1}^{N} b_{\ell } \leqslant b_N ( 1 + r + \cdots + r^{N -N' -1} ) \\\leqslant & {} b_N \frac{ r^{N- N'}}{ r -1 } \leqslant b_N \frac{r^N}{r-1} = \frac{C'}{r-1} (\epsilon k )^{-N} N! \end{aligned}$$

which concludes the proof. $\square$

Let us call a formal series $a = \sum \hbar ^\ell a_{\ell }$ of $E[[\hbar ]]$ an analytic symbol if $|a_{\ell }| \leqslant C^{\ell +1} \ell !$ for some $C>0$. For any $\epsilon >0$, we set $a(\epsilon ,k) := \sum _{\ell =0 }^{\lfloor \epsilon k \rfloor } k^{-\ell } a_\ell$. Choose a second normed space $(E', | \cdot | ')$ and let $\mathcal {L} ( E, E')$ (resp. $\mathcal {B} (E,E')$) be the space of bounded linear maps $E \rightarrow E'$ (resp. bounded bilinear maps $E \times E \rightarrow E'$) with its natural norm.

Lemma 5.4

• 1. For any analytic symbols $a \in E[[\hbar ]]$ and $P \in \mathcal {L} (E, E') [[\hbar ]]$, the series $b = \sum _{\ell , m } \hbar ^{\ell + m } P_{\ell } ( a_m)$ is an analytic symbol of $E'[[\hbar ]]$. Furthermore, if $\epsilon >0$ is sufficiently small, then there exists $C>0$ such that

  $$\begin{aligned} P(\epsilon ,k ) (a ( \epsilon ,k ) ) = b ( \epsilon ,k) + \mathcal {O}( e^{- k/C} ). \end{aligned}$$

• 2. For any analytic symbols $a, a' \in E[[\hbar ]]$ and $B \in \mathcal {B} (E, E') [[\hbar ]]$, the series $b = \sum _{\ell , m,p } \hbar ^{\ell + m+p } B_{\ell } ( a_m, a'_p)$ is an analytic symbol of $E'[[\hbar ]]$. Furthermore if $\epsilon >0$ is sufficiently small, then there exists $C>0$ such that

  $$\begin{aligned} B(\epsilon ,k ) (a ( \epsilon ,k ), a' ( \epsilon , k) ) = b ( \epsilon ,k) + \mathcal {O}( e^{- k/C} ). \end{aligned}$$

Proof

Assume that $\Vert P_{\ell } \Vert \leqslant C^{\ell +1} \ell !$ and $| a_{m} | \leqslant C^{m+1} m!$. Then $| P_{\ell } a_{m} | \leqslant C^2 C^{\ell +m } (\ell + m )!$, so $| b_p | \leqslant (p+1) C^{2 + p} p!$ and we conclude that b is analytic. Furthermore, for $N = \lfloor \epsilon k \rfloor$, we have

$$\begin{aligned} \bigl | P(\epsilon ,k ) (a ( \epsilon ,k ) ) - b ( \epsilon ,k) \bigr | \leqslant \sum ^{\ell \leqslant N, \; m \leqslant N}_{N< \ell + m} k^{-\ell -m} |P_\ell (a_m)| \leqslant N \sum _{p = N+1}^{2N} C^2 (Cp/k)^p \end{aligned}$$

by the previous estimate. $N < p\leqslant 2N$ implies that $\epsilon k \leqslant p \leqslant 2 \epsilon k$ so that $(Cp/k)^p \leqslant (2 C \epsilon )^p \leqslant (2 C\epsilon )^{\epsilon k}$ where we have assumed that $2C \epsilon <1$. It follows that

$$\begin{aligned} \bigl | P(\epsilon ,k ) (a ( \epsilon ,k ) ) - b ( \epsilon ,k) \bigr | \leqslant N^2 C^2 (2 C \epsilon )^{\epsilon k} \leqslant (\epsilon C) ^2 k^2 (2 C\epsilon )^{\epsilon k} = \mathcal {O}(e^{-k/C'}) \end{aligned}$$

for a sufficiently large $C'$. The proof of the second part is similar. $\square$

1.2 Estimates for holomorphic star-products

Let ${\text {End}} (E)$ be the algebra of endomorphisms of a vector space E. Let $(\Vert \cdot \Vert _\ell , \ell \in \mathbb {N})$ be a family of seminorms of ${\text {End}} E$ satisfying

$$\begin{aligned} \Vert P \circ Q \Vert _{p+q} \leqslant \Vert P \Vert _p \Vert Q\Vert _q , \qquad \forall P, Q \in {\text {End}} E \end{aligned}$$

(36)

for any $p, q \in \mathbb {N}$. Consider a formal series ${\text {id}} - \sum _{\ell \geqslant 1} \hbar ^{\ell } F_\ell$ of $({\text {End}} E)[[\hbar ]]$ with inverse ${\text {id}} + \sum _{\ell \geqslant 1} \hbar ^{\ell } G_\ell$.

Lemma 5.5

If there exists $C>0$ such that $\Vert F_{\ell } \Vert _{\ell } \leqslant C^{\ell }$ for any $\ell$, then there exists $C'>0$ such that $\Vert G_{\ell } \Vert _{\ell } \leqslant (C')^{\ell }$ for any $\ell$.

We give two proofs, the first one is a direct generalization of the proof of Theorem 3.1, the second is inspired from [5, 16].

Proof

A first proof is to establish the formula

$$\begin{aligned} G_{m} = \sum _{\begin{array}{c} \ell \geqslant 1, \; (i_1, \ldots , i_{\ell }) \in \mathbb {Z} _{>0}^\ell ,\\ i_1 + \cdots + i_\ell = m \end{array} } F_{i_1} \ldots F_{i_{\ell }} \end{aligned}$$

(37)

and we conclude easily by using that the number of terms in the sum is $2^{m-1}$. Another proof less precise but interesting as well is to introduce for any formal series $R= \sum \hbar ^\ell R_{\ell }$ and $\rho >0$ the series $f( R, \rho ) = \sum \rho ^{\ell } \Vert R_{\ell } \Vert _{\ell }$. Then

• 1. $f(R,\rho )$ converges for some $\rho >0$ if and only there exists $C>0$ such that $\Vert R_{\ell } \Vert _{\ell } \leqslant C^{\ell +1}$ for any $\ell$.

• 2. $f(RS, \rho ) \leqslant f( R, \rho ) f(S, \rho )$ by (36).

• 3. if $(R^{(n)})_n$ is a sequence of $({\text {End}} E)[[\hbar ]]$ such that $R^{(n)} = \mathcal {O}( \hbar ^n)$ for any n, then $f ( \sum _n R^{(n)} , \rho ) \leqslant \sum _n f(R^{(n)} , \rho )$ by triangle inequality.

Then we can argue as follows. Set $R = \sum _{\ell \geqslant 1} \hbar ^{\ell } F_{\ell }$. The assumption on the $F_{\ell }$’s implies that $f ( R, \rho ) = \mathcal {O}( \rho )$, so $f( R, \rho ) \leqslant \delta <1$ when $\rho$ is sufficiently small. Applying the previous properties we have

$$\begin{aligned} f( \sum R^n, \rho ) \leqslant \sum f( R^n, \rho ) \leqslant \sum \delta ^n < \infty \end{aligned}$$

which concludes the proof because ${\text {id}} + \sum \hbar ^{\ell } G_{\ell } = \sum R^n$. $\square$

We apply this to holomorphic differential operators of an open set $\Omega$ of $\mathbb {C} ^n$. Consider a formal series ${\text {id}} - \sum _{\ell \geqslant 1} \hbar ^{\ell } F_\ell$ where for any $\ell$,

$$\begin{aligned} F_{\ell } = \sum _{|\alpha |\leqslant N \ell } a_{\ell , \alpha } \frac{1}{\alpha !} \partial ^\alpha \end{aligned}$$

and the $a_{\alpha , \ell }$’s are holomorphic functions of $\Omega$. Here N is any positive integer, $N=1$ or 2 in our applications. Then the inverse ${\text {id}} + \sum _{\ell \geqslant 1} \hbar ^{\ell } G_\ell$ has the same form $G_{\ell } = \sum _{|\alpha |\leqslant N \ell } b_{\ell , \alpha } \frac{1}{\alpha !} \partial ^\alpha$ as follows for instance from (37).

Lemma 5.6

Assume that on any compact set K of $\Omega$, there exists $C_K$ such that $|a_{\ell , \alpha } | \leqslant C_K^{\ell } \ell ^{\ell }$ for any $\alpha$, $\ell$. Then the family $(b_{\ell , \alpha })$ satisfies the same estimates, with different constants $C_K$.

Proof

Let $x_0 \in \Omega$ and $0<t_0\leqslant 1$ be such that the closure of the ball $B(x_0, t_0)$ is contained in $\Omega$. For any $0<s<t < t_0$, define as in Sect. 2.3$\Vert P \Vert _{t,s}$ as the norm of the restriction $P : \mathcal {B} ( B(x_0, t)) \rightarrow \mathcal {B} ( B(x_0, s))$. Set

$$\begin{aligned} \Vert P \Vert _{\ell } := \sup \{ \ell ^{-\ell } (t-s)^\ell \Vert P \Vert _{t,s} / \; 0<s<t<t_0 \} . \end{aligned}$$

The submultiplicativity (36) follows from Lemma 2.4. By the first part of Lemma 2.4 and the assumption on the $a_{\ell ,\alpha }$’s, we have $\Vert F_{\ell } \Vert _{\ell } \leqslant C^\ell$. This implies by Lemma 5.5 that $\Vert G_{\ell } \Vert _{\ell } \leqslant C^\ell$ with a larger C. For any $x \in B (x_0, t_0/2)$, the sup norm of $(z - x)^\alpha$ on $B(x_0,t_0)$ is smaller than $(3 t_0/2)^{|\alpha |} \leqslant (3/2)^{\ell }$. So

$$\begin{aligned} | b_{\ell , \alpha } (x) | = | G_{\ell } ( (z -x )^{\alpha } ) (x) |\leqslant & {} (3/2)^{\ell } \Vert G_{\ell } \Vert _{t_0, \frac{t_0}{2}} \\\leqslant & {} (3/2)^{\ell } \ell ^{\ell } (t_0/2)^{-\ell } \Vert G_{\ell } \Vert _{\ell } \leqslant (3C /t_0)^{\ell } \ell ^{\ell } . \end{aligned}$$

as was to be proved. $\square$

A first application is the proof of the claim in Remark 5.2. It is also easy to recover Theorem 3.1. A third application is to prove that the inverse for the product $\star$ of an analytic elliptic symbol $f =\sum \hbar ^{\ell } f_{\ell }$ is analytic. Here, elliptic means that $f_0$ does not vanish anywhere, and thus has an inverse $f_0^{-1}$ for the usual product. The inverse g of f satisfies $f \star g = \rho$ where $\rho$ is the Bergman kernel symbol. So $(f_0^{-1} \star f ) \star g = f_0^{-1} \star \rho$. Now $(f_0^{-1} \star f) \star g = ({\text {id}} - \sum \hbar ^{\ell } F_{\ell }) (g)$ and Theorem 2.3 implies that the coefficients of the $F_{\ell }$’s satisfy the assumption of Lemma 5.6. On the other hand $f_0^{-1} \star \rho$ is analytic by Proposition 4.1. Thus $g = ( {\text {id}} + \sum \hbar ^{\ell } G_{\ell } ) ( f_0^{-1} \star \rho )$ is analytic by Lemma 5.6.