Off-Diagonal Berezin Symbol

• Off-diagonal Berezin symbol is a two-point normalized kernel in reproducing-kernel Hilbert spaces that extends the diagonal Berezin symbol to pairs of points.
• It is computed by normalizing an operator's matrix elements with the reproducing kernel, offering a practical tool for symbolic calculus in quantization frameworks.
• Its explicit Gaussian structure in models such as the Fock–Bargmann space aids in deriving Weyl symbols and analyzing metaplectic operators.

Searching arXiv for the cited papers to ground the response. The off-diagonal Berezin symbol is a two-point symbol attached to an operator on a reproducing-kernel Hilbert space, obtained by evaluating the operator between two coherent states and normalizing by their overlap. In the setting of holomorphic quantization, it extends the usual diagonal Berezin symbol from one point to a pair of points, and in the paper “Complex Weyl symbols of metaplectic operators: an elementary approach” it appears under the name double Berezin symbol (Cahen, 2023). In that framework, as in several related representation-theoretic and Berezin-type calculi, the off-diagonal symbol is the normalized integral kernel of the operator, and the diagonal symbol is recovered by restriction to the diagonal (Cahen, 2023).

1. Definition and basic structure

In the Fock–Bargmann model of (Cahen, 2023), for $\lambda>0$ the Fock space $F_\lambda$ consists of holomorphic functions $f:\mathbb C^n\to\mathbb C$ such that

$$\|f\|^2_{F_\lambda}=\int_{\mathbb C^n}|f(z)|^2e^{-\lambda|z|^2/2}\,d\mu_\lambda(z)<+\infty,$$

with

$d\mu_\lambda(z):=(\lambda/2\pi)^n\,dm(z).$

Its coherent states are

$$e_z(w)=\exp\!\left(\frac{\lambda}{2}\,z w\right),\qquad w\in\mathbb C^n,$$

and satisfy the reproducing property

$f(z)=\langle f,e_z\rangle_{F_\lambda}.$

For a bounded operator $A$ on $F_\lambda$, the diagonal Berezin symbol is

$$S_\lambda(A)(z)=\frac{\langle A e_z,e_z\rangle_{F_\lambda}}{\langle e_z,e_z\rangle_{F_\lambda}}.$$

The off-diagonal Berezin symbol, called the double Berezin symbol in (Cahen, 2023), is

$F_\lambda$0

It is holomorphic in $F_\lambda$1 and anti-holomorphic in $F_\lambda$2, and its diagonal restriction recovers the ordinary Berezin symbol: $F_\lambda$3

The same structural definition appears in a broader form in reproducing-kernel Hilbert spaces associated with Lie-group representations. In “Berezin symbols on Lie groups,” the full symbol of an operator $F_\lambda$4 is

$F_\lambda$5

and the diagonal Berezin covariant symbol is its restriction $F_\lambda$6 (Beltita et al., 2016). This identifies the off-diagonal Berezin symbol with the full two-point kernel, while the diagonal symbol is a restriction.

2. Kernel-theoretic interpretation

The defining structural relation in (Cahen, 2023) is

$F_\lambda$7

Thus the off-diagonal Berezin symbol is the operator kernel divided by the reproducing kernel. In this sense it is literally a normalized kernel.

Given $F_\lambda$8, the action of $F_\lambda$9 is recovered by

$f:\mathbb C^n\to\mathbb C$0

The same pattern persists in the holomorphic representation spaces $f:\mathbb C^n\to\mathbb C$1 attached to the Jacobi group. There the reproducing kernel is $f:\mathbb C^n\to\mathbb C$2, the coherent states are $f:\mathbb C^n\to\mathbb C$3, and the off-diagonal symbol is

$f:\mathbb C^n\to\mathbb C$4

This suggests that, across these holomorphic settings, the off-diagonal Berezin symbol is the most intrinsic kernel-level object, while the diagonal symbol is a derived quantity (Cahen, 2023).

A closely related viewpoint is explicit in “Berezin symbols and spectral measures of representation operators,” where the off-diagonal object is called the double Berezin symbol: $f:\mathbb C^n\to\mathbb C$5 There too, the operator kernel is

$f:\mathbb C^n\to\mathbb C$6

with $f:\mathbb C^n\to\mathbb C$7 the reproducing kernel (Cahen, 2020).

3. Fock–Bargmann and Jacobi-group realizations

The paper (Cahen, 2023) treats the off-diagonal symbol in two linked settings.

First is the standard Fock–Bargmann space $f:\mathbb C^n\to\mathbb C$8 for the Heisenberg group, where the coherent states $f:\mathbb C^n\to\mathbb C$9 and their overlaps are explicit. In this model,

$$\|f\|^2_{F_\lambda}=\int_{\mathbb C^n}|f(z)|^2e^{-\lambda|z|^2/2}\,d\mu_\lambda(z)<+\infty,$$0

The off-diagonal symbol is therefore an explicit ratio of two Gaussian-type kernels.

Second are the holomorphic representation spaces $$\|f\|^2_{F_\lambda}=\int_{\mathbb C^n}|f(z)|^2e^{-\lambda|z|^2/2}\,d\mu_\lambda(z)<+\infty,$$1 attached to the Jacobi group

$$\|f\|^2_{F_\lambda}=\int_{\mathbb C^n}|f(z)|^2e^{-\lambda|z|^2/2}\,d\mu_\lambda(z)<+\infty,$$2

realized on a bounded symmetric domain

$$\|f\|^2_{F_\lambda}=\int_{\mathbb C^n}|f(z)|^2e^{-\lambda|z|^2/2}\,d\mu_\lambda(z)<+\infty,$$3

The Jacobi representation acts by

$$\|f\|^2_{F_\lambda}=\int_{\mathbb C^n}|f(z)|^2e^{-\lambda|z|^2/2}\,d\mu_\lambda(z)<+\infty,$$4

with coherent states again given by the reproducing kernel. The metaplectic representation on $$\|f\|^2_{F_\lambda}=\int_{\mathbb C^n}|f(z)|^2e^{-\lambda|z|^2/2}\,d\mu_\lambda(z)<+\infty,$$5 is then obtained by restricting the Jacobi-group kernel to the submanifold $$\|f\|^2_{F_\lambda}=\int_{\mathbb C^n}|f(z)|^2e^{-\lambda|z|^2/2}\,d\mu_\lambda(z)<+\infty,$$6, namely

$$\|f\|^2_{F_\lambda}=\int_{\mathbb C^n}|f(z)|^2e^{-\lambda|z|^2/2}\,d\mu_\lambda(z)<+\infty,$$7

This restriction mechanism is central: it transfers explicit kernel formulas from the Jacobi model to the Fock model, and hence transfers explicit off-diagonal Berezin symbols for metaplectic operators (Cahen, 2023).

A plausible implication is that the Jacobi-group realization provides a uniform source of kernel formulas, while the Fock realization is the computationally simplest locus for extracting concrete off-diagonal symbols.

4. Explicit Gaussian formula for metaplectic operators

For

$$\|f\|^2_{F_\lambda}=\int_{\mathbb C^n}|f(z)|^2e^{-\lambda|z|^2/2}\,d\mu_\lambda(z)<+\infty,$$8

the Fock-space kernel of the metaplectic operator $$\|f\|^2_{F_\lambda}=\int_{\mathbb C^n}|f(z)|^2e^{-\lambda|z|^2/2}\,d\mu_\lambda(z)<+\infty,$$9 is

$d\mu_\lambda(z):=(\lambda/2\pi)^n\,dm(z).$0

Dividing by the reproducing kernel yields the off-diagonal Berezin symbol

$d\mu_\lambda(z):=(\lambda/2\pi)^n\,dm(z).$1

This is the central explicit formula of (Cahen, 2023). It is Gaussian in the two-point variables and reduces to the diagonal symbol when $d\mu_\lambda(z):=(\lambda/2\pi)^n\,dm(z).$2.

The corresponding diagonal Berezin symbol is

$d\mu_\lambda(z):=(\lambda/2\pi)^n\,dm(z).$3

Hence the off-diagonal symbol is not merely a technical extension of the diagonal one: it is the more primitive Gaussian kernel formula from which the diagonal expression is obtained by restriction.

For infinitesimal generators $d\mu_\lambda(z):=(\lambda/2\pi)^n\,dm(z).$4, (Cahen, 2023) gives the diagonal symbol

$d\mu_\lambda(z):=(\lambda/2\pi)^n\,dm(z).$5

and states that the off-diagonal symbol is obtained by computing $d\mu_\lambda(z):=(\lambda/2\pi)^n\,dm(z).$6 and dividing by $d\mu_\lambda(z):=(\lambda/2\pi)^n\,dm(z).$7, structurally yielding a quadratic form in $d\mu_\lambda(z):=(\lambda/2\pi)^n\,dm(z).$8 and $d\mu_\lambda(z):=(\lambda/2\pi)^n\,dm(z).$9. This suggests that the Gaussian two-point structure persists infinitesimally.

5. Relation to Weyl symbols and quantization formalisms

A major theme of (Cahen, 2023) is the passage from off-diagonal kernels to complex Weyl symbols. The complex Weyl symbol $$e_z(w)=\exp\!\left(\frac{\lambda}{2}\,z w\right),\qquad w\in\mathbb C^n,$$0 is defined by

$$e_z(w)=\exp\!\left(\frac{\lambda}{2}\,z w\right),\qquad w\in\mathbb C^n,$$1

and has the integral representation

$$e_z(w)=\exp\!\left(\frac{\lambda}{2}\,z w\right),\qquad w\in\mathbb C^n,$$2

Thus the complex Weyl symbol is obtained from the off-diagonal kernel $$e_z(w)=\exp\!\left(\frac{\lambda}{2}\,z w\right),\qquad w\in\mathbb C^n,$$3, hence from the off-diagonal Berezin symbol, by a Gaussian integral transform. The paper also states

$$e_z(w)=\exp\!\left(\frac{\lambda}{2}\,z w\right),\qquad w\in\mathbb C^n,$$4

where $$e_z(w)=\exp\!\left(\frac{\lambda}{2}\,z w\right),\qquad w\in\mathbb C^n,$$5 is the Berezin transform. In this formulation, the diagonal Berezin symbol is derived from the Weyl symbol, but the Weyl symbol itself is computed through off-diagonal kernel data (Cahen, 2023).

The same paper then relates $$e_z(w)=\exp\!\left(\frac{\lambda}{2}\,z w\right),\qquad w\in\mathbb C^n,$$6 to the classical Weyl symbol $$e_z(w)=\exp\!\left(\frac{\lambda}{2}\,z w\right),\qquad w\in\mathbb C^n,$$7 via the Bargmann transform, recovering known formulas for metaplectic operators and, in particular, the Weyl symbol of the exponential of an operator whose Weyl symbol is a quadratic form.

More broadly, several papers in the supplied corpus treat off-diagonal kernel data as the operative object behind symbolic calculus. In the compact symplectic Berezin–Toeplitz setting, “On the composition of Berezin-Toeplitz operators on symplectic manifolds” computes composition coefficients from the full off-diagonal expansion of the Bergman kernel (Ioos, 2017). In the finite-regularity Toeplitz setting, “Semi-classical properties of Berezin–Toeplitz operators with $$e_z(w)=\exp\!\left(\frac{\lambda}{2}\,z w\right),\qquad w\in\mathbb C^n,$$8-symbol” identifies the off-diagonal Berezin symbol with the normalized Toeplitz kernel and shows that near the diagonal it is asymptotically the classical observable (Barron et al., 2013). These works reinforce the same pattern: diagonal symbols encode observables, but off-diagonal kernels drive composition and asymptotics.

6. Generalizations, terminology, and scope

The terminology is not uniform across the literature. In (Cahen, 2023) and (Cahen, 2020), the expression “double Berezin symbol” is used for the two-point normalized kernel. In (Beltita et al., 2016), the analogous object is the full symbol $$e_z(w)=\exp\!\left(\frac{\lambda}{2}\,z w\right),\qquad w\in\mathbb C^n,$$9, and the diagonal Berezin symbol is the restriction $f(z)=\langle f,e_z\rangle_{F_\lambda}.$0. In “Berezin-Type Operators on the Cotangent Bundle of a Nilpotent Group,” the two-point covariant symbol

$f(z)=\langle f,e_z\rangle_{F_\lambda}.$1

plays the same role (Mantoiu, 2019). In “Berezin quantization and representation theory,” the Berezin transform is written with an explicitly off-diagonal kernel

$f(z)=\langle f,e_z\rangle_{F_\lambda}.$2

again showing that the calculus is built from two-point data (Molchanov, 2023).

By contrast, some works study only the diagonal transform. “Toeplitz algebra and Symbol map via Berezin transform on $f(z)=\langle f,e_z\rangle_{F_\lambda}.$3” defines

$f(z)=\langle f,e_z\rangle_{F_\lambda}.$4

and explicitly states that it does not introduce an off-diagonal Berezin transform $f(z)=\langle f,e_z\rangle_{F_\lambda}.$5 (Javed et al., 2024). This is a useful corrective to a common misconception: off-diagonal Berezin symbols are natural in reproducing-kernel settings, but they are not automatic ingredients of every Berezin-transform-based analysis.

Another misconception is that the off-diagonal symbol is merely redundant because it is determined by the diagonal one. In (Cahen, 2023), analyticity implies that the two-point function is determined by its diagonal restriction in the Fock setting. But the same paper also shows that the explicit computation of metaplectic and Weyl symbols proceeds through kernels and two-point formulas. This suggests that even where diagonal data determine the two-point function abstractly, off-diagonal formulas remain the practical vehicle for symbolic calculus.

In summary, the off-diagonal Berezin symbol is best understood as the normalized two-point kernel of an operator in a reproducing-kernel quantization scheme. In the metaplectic setting of (Cahen, 2023), it takes an explicit Gaussian form; in Lie-group and Toeplitz settings, it appears as the full covariant kernel; and across these contexts it mediates between coherent-state matrix coefficients, integral kernels, Berezin transforms, and Weyl-type symbol calculi (Cahen, 2023).