Bivariate Berezin Transform

• The bivariate Berezin transform is a generalization that uses pairs of reproducing kernels to extend classical operator symbol calculus.
• It employs coherent state constructions and kernel factorizations, enabling refined analysis of off-diagonal behavior in complex and noncommutative domains.
• Applications span quantization, spectral analysis, and operator theory, providing deeper insights into function spaces and multidimensional harmonic analysis.

The bivariate Berezin transform extends the classical Berezin transform to settings involving two variables or points, typically in reproducing kernel Hilbert spaces associated to domains of several complex variables, noncommutative polydomains, or phase spaces with additional symmetry. This generalization facilitates a more refined symbolic calculus, allowing for the analysis of operator correlations, off-diagonal behavior, and interactions between paired observables or data points. The construction, properties, and applications of the bivariate Berezin transform are diverse, influencing quantization, operator theory, spectral analysis, and harmonic analysis.

1. Foundational Definition and Kernel Structure

The classical Berezin transform on a function space $\mathcal{H}$ (e.g., Bergman or Hardy space) uses reproducing kernels $K_z$ to associate operators $T$ to functions via

$\tilde{T}(z) = \langle T k_z, k_z \rangle$

where $k_z$ is the normalized kernel at $z$. The bivariate Berezin transform arises by evaluating $T$ on pairs of normalized kernels:

$\mathcal{B}_T(z, w) := (T(k_z), k_w)_{A^2}$

or by integrating a symbol $u$ against a bivariate kernel:

$$B(u)(z, w) = \int_\Omega u(\zeta) K(z, \zeta) \overline{K(w, \zeta)}\, d\mu(\zeta)$$

On compact Kähler manifolds and certain homogeneous spaces, the kernel can be written using coherent states or Bergman kernels as $B^{(m)}(a, b)B^{(m)}(b, a)$, leading to integral representations where off-diagonal properties become prominent (Schlichenmaier, 2010).

In noncommutative, polydomain, or para-Hermitian settings, the Berezin kernel may be explicitly constructed in terms of symmetry and intertwining operators, as in

$$B(\xi, \eta; u, v) = c \frac{\Phi(\xi, v) \Phi(u, \eta)}{\Phi(\xi, \eta) \Phi(u, v)}$$

where $\Phi$ encodes the action of the group and reproducing kernel structure (Molchanov, 2023).

2. Symbolic Calculus and Function Spaces

The bivariate Berezin transform bridges contravariant and covariant symbols by lifting functions or operators to symbol spaces defined on $M \times M$ (e.g., the product of a domain with itself). In quantization contexts, the transform "corrects" the Toeplitz operator symbol to yield the covariant symbol via coherent states:

$$I^{(m)}(f)(x) = \langle e^{(m)}(x), T^{(m)}(f) e^{(m)}(x) \rangle$$

with the integral form

$$I^{(m)}(f)(x) = \frac{1}{u_m(x)} \int_M u_m(x, y) f(y)\, d\mu(y)$$

where $u_m(x, y)$ is the bivariate kernel (Schlichenmaier, 2010).

For domains with product structure, such as polydiscs or pseudoconvex product domains, the reproducing kernel factors:

$$k_z(w) = k_{z_1}(w_1) k_{z_2}(w_2) \cdots k_{z_m}(w_m)$$

and the Berezin transform for products of operators decomposes accordingly:

$$\tilde{T}(z) = \tilde{T}_1(z_1) \cdot \tilde{T}_2(z_2) \cdots \tilde{T}_m(z_m)$$

This separation facilitates the analysis of regularity and norm estimates across factors (Cuckovic et al., 2019).

3. Noncommutative, Multivariate, and Bivariate Extensions

The bivariate Berezin transform generalizes naturally to noncommutative settings. In polydomains, the kernel is constructed for operator tuples, and the transform becomes (Popescu, 2013):

$$B_{T, J}[g] = K_{f, T, J}^* (g \otimes I_{\mathcal{H}}) K_{f, T, J}$$

where $K_{f, T, J}$ is a constrained Berezin kernel on a quotient space encoding the variety's relations.

In Hardy algebras over $W^*$-correspondences, the Berezin transform maps elements to analytic operator-valued functions on domains with "matricial structure." The bivariate extension appears as a transform between correspondence duals, retaining intertwining properties and tensorial expansions, but now with coefficients that are bimodule maps (Muhly et al., 2012).

For time-frequency analysis, the Berezin transform in phase space acts on operators $T$ via window functions $\varphi_1, \varphi_2$:

$$\mathcal{B}T(z) = \langle T \pi(z)\varphi_2, \pi(z)\varphi_1 \rangle$$

This bivariate structure (in time and frequency) reflects non-holomorphic domains and quantization schemes unrelated to complex geometry (Bayer et al., 2014). The injectivity and density results in Schatten class operator spaces hinge on the nonvanishing of certain time-frequency overlaps.

4. Weighted Inequalities and Function Theory

In weighted Bergman and Hardy spaces, the bivariate Berezin transform inherits the sharp weighted estimates present in the univariate case. For the unit ball in $\mathbb{C}^n$, the Berezin transform is (Rahm et al., 2016):

$$B_b f(z) = (1 - |z|^2)^{n+1+b} \int_{B_n} f(w) |1 - z\overline{w}|^{-(2n+2+2b)} dv_b(w)$$

For bivariate settings (e.g., functions on $B_n \times B_n$), the dyadic harmonic analysis framework can be adapted, with weight characteristics extended to product-type norms.

On weighted Bergman spaces induced by regular weights, boundedness, compactness, and Schatten class membership of Toeplitz operators can be characterized via the Berezin transform and appropriate averaged Carleson measures. The bivariate version is achieved by evaluating $T$ on pairs of normalized kernels (Peláez et al., 2016).

5. Spectral, Geometric, and Quantization Aspects

Spectral properties of the bivariate Berezin transform connect directly to geometric quantization and Markov processes. In Kähler manifolds, the Berezin transform admits an asymptotic expansion involving the Laplace–Beltrami operator:

$B(f) \approx f - \frac{\Delta}{4T} f + O(p^{-2})$

with kernel-level representations involving the squared modulus of Bergman kernels. The spectral gap is determined by the fundamental tone, confirming conjectures in complex geometry and balanced metrics (Ioos et al., 2018).

In homogeneous spaces (e.g., para-Hermitian symmetric spaces or complex projective spaces), the Berezin transform is constructed as a function of invariant Laplacians, often via coherent states and spectral analysis. Variational formulas explicitly link the transform to Laplacian eigenstates and joint invariants, and the transform's kernel can be adapted to bivariate forms for more general quantization schemes (Demni et al., 2016, Askour, 2017, Molchanov, 2023).

6. Applications and Operator-Theoretic Implications

Operator theory: The bivariate Berezin transform enables the paper of Toeplitz algebra homomorphisms, symbol maps, and compactness criteria, in both scalar-valued and vector-valued spaces. The transform extracts the symbol of an operator, particularly in multivariate Hardy and Bergman spaces, using radial or boundary limits, and determines essential norm equivalence on product domains (Javed et al., 1 May 2024, Rahm, 2014).

Function-theoretic range: Decomposition results such as those of Ahern, Čučković, and Li apply, with finite-rank Berezin transforms characterized by product expansions over holomorphic symbols and automorphisms. The bivariate setting accommodates Taylor expansions, finite-dimensional linear structures, and distribution-theoretic arguments, supporting the resolution of symbol calculus and zero-product problems (Rao, 2010).

Quantization and deformation: The bivariate Berezin transform underpins strict quantization, star product classification, and separation-of-variables properties on Kähler and homogeneous spaces, bridging classical observables and operator symbols via its kernel representation (Schlichenmaier, 2010, Molchanov, 2023).

7. Perspectives and Research Directions

The extension of the Berezin transform to bivariate and multivariate contexts is robust in theory and tools:

• Techniques from dyadic harmonic analysis, spectral decomposition, and functional calculus can be generalized for product domains and multidimensional settings.
• The interplay between boundary regularity, kernel factorization, and symbolic mappings governs compactness and norm estimates of operator classes.
• In noncommutative and operator-valued settings, the fully matricial structure and intertwining properties enable tensorial expansions and facilitate the paper of linear matrix inequalities and free analysis.

A plausible implication is that bivariate Berezin transforms will continue to provide new insights in geometric quantization, operator algebra theory, spectral geometry, and harmonic analysis, particularly in domains that admit rich symmetry or product structure. The explicit kernel constructions, norm estimates, and symbolic decompositions form a foundation for further exploration in higher-dimensional and noncommutative function theory.