Berezin Number Inequalities in Operator Theory

• Berezin number inequalities define and bound the maximal modulus of the Berezin transform for bounded linear operators in reproducing kernel Hilbert spaces.
• They compare Berezin–sectorial operators with classical sectorial ones, elucidating differences in spectral and geometric properties.
• Key results include sine bounds, mixed power inequalities, and applications to Toeplitz operators in spaces like the Bergman and Hardy spaces.

The Berezin number of a bounded linear operator on a reproducing-kernel Hilbert space is a metric that captures the maximal modulus of the Berezin transform, which itself is defined by testing the operator against normalized kernel functions. Inequalities involving the Berezin number are central to spectral and geometric aspects of operator theory, particularly when considering sectoriality relative to distinguished sectors in the complex plane. Recent advances generalize classical sectorial operator theory, introducing the notion of Berezin sectorial operators and refining related inequality bounds for broad classes of function-theoretic operators (Mahapatra et al., 6 Jan 2026).

1. Berezin Transform, Berezin Range, and Berezin Number

Let $\mathcal{H}$ be a reproducing-kernel Hilbert space over a set $\Omega$, with normalized kernels $\{\hat k_z : z\in \Omega\}$. For $T\in B(\mathcal{H})$, the Berezin transform is

$$\widetilde T(z) = \langle T\hat k_z, \hat k_z \rangle, \quad z\in \Omega.$$

The Berezin range is the image

$\Ber(T) = \{\widetilde T(z): z\in\Omega\},$

and the Berezin number is the maximal modulus of the Berezin transform: $\ber(T) = \sup_{z\in\Omega} |\widetilde T(z)|.$ For any $\theta\in [0,\pi/2)$, the closed sector $S_\theta$ is defined by

$$S_\theta = \{ w\in\mathbb C: |\arg w| \leq \theta\}.$$

The operator $T$ is called Berezin–sectorial of angle $\theta$ (notation: $T\in\Pi_\theta^{\rm Ber}$) if $\Ber(T)\subseteq S_\theta$, or equivalently $|\arg \widetilde T(z)|\le \theta$ for all $z$.

2. Comparison with Classical Sectoriality

Classical sectoriality of a bounded Hilbert space operator $T$ requires its numerical range

$W(T) = \{\langle T x, x\rangle: \|x\|=1\}$

to be contained in a sector $S_\theta$. Every sectorial operator in the classical sense is Berezin–sectorial, since $\Ber(T)\subseteq W(T)$. However, this inclusion can be strict: there exist operators that are Berezin–sectorial but not classically sectorial.

For example, consider the Hardy space $H^2(\mathbb D)$ with $\phi(z)=\rho z$, $0<\rho<1$, and the composition-differentiation operator $D_\phi(f)=f'\circ\phi$. The Berezin transform evaluates to

$$\widetilde{D_\phi}(re^{i\zeta}) = (1-r^2)\,\frac{\rho\,r}{(1-\rho\,r^2)^2} e^{i\zeta}, \quad 0\le r<1,$$

so $\Ber(D_\phi)$ is the closed disk $\{w\in\mathbb C : |w|\le r_1(\rho)\}$ for certain $r_1(\rho)$. Meanwhile, the numerical range $W(D_\phi)$ is a disk of generally larger radius. By shifting $D_\phi$ by appropriate scalars, one can construct examples where the operator is Berezin–sectorial of smaller angle than any possible classical sectorial angle.

This demonstrates that the Berezin–sectorial index—$\sup|\arg \widetilde T(z)|$—can be strictly smaller than the classical sectorial index $\sup|\arg \langle Tx,x\rangle|$.

3. Berezin Number Inequalities for Sectorial Operators

For $T\in\Pi^{\rm Ber}_\theta$ with Cartesian decomposition $T=\Re T + i\,\Im T$, several sharp inequalities hold:

• Sine bound for the imaginary part:

$\sin\theta\,\ber(T)\ge \ber(\Im T).$

• Lower bound in terms of real and imaginary parts (for $\theta\ne 0$):

$\ber(T)\;\ge\; \frac{\csc\theta}{2}\,\ber(\Re T\pm\Im T) + \frac{\csc\theta}{2}\left[\ber(\Im T)-\ber(\Re T)\right].$

• Two-sided sector variant: If $\Ber(T)\subset \{re^{-i\phi}: \phi_1\le\phi\le\phi_2\}$ and $\phi_2<\pi/2$, then the sharp bound uses $\theta=\max\{\phi_2,\tfrac\pi2-\phi_1\}$.
• Product lower bound: If $T^*S\in\Pi^{\rm Ber}_\theta$ for some $S$, then for any $\alpha>0$,

$\ber(T^*S)\ge\max\{\beta_1,\beta_2\},$

with

$$\beta_1 = \frac{\csc\theta}{2\alpha}\left(\|\,|S+i\alpha T|^2\|_{ber} - \|S^*S+\alpha^2T^*T\|_{ber}\right),$$

$$\beta_2 = \frac{\csc\theta}{2\alpha}\left(\|S^*S+\alpha^2T^*T\|_{ber} - \|\,|S-i\alpha T|^2\|_{ber}\right).$$

• Invertible factor bound: If $S$ invertible, $S^*T\in\Pi^{\rm Ber}_\theta$,

$\|\,|T|^2\|_{ber}^{1/2} \le \|S^{-1}\|\left[\sin\theta\,\ber(S^*T) + \frac12\,\|\,|T-iS|^2\|_{ber}\right].$

• Mixed power inequality: For all real $t$,

$\|\,|T|^2\|_{ber} \le \frac14(\sin\theta+1)^2\ber^2(T) + \frac12\inf_{t\in\mathbb R}\left(\|\,|T-tI|^2\|_{ber}+\|\,|T-itI|^2\|_{ber}\right).$

Additionally, for Toeplitz operators $T_\phi$ on the weighted Bergman space $A^2_\alpha(\mathbb D)$ with $\phi$ harmonic, $\ber(T_\phi^n)\le \ber^n(T_\phi)$ holds for all $n$.

A subclass $\Pi^{\rm Ber,P}_\theta$ of $\Pi^{\rm Ber}_\theta$ is defined by also requiring $\ber(\Re^nT)\le\ber^n(\Re T)$ and $\ber(\Im^nT)\le\ber^n(\Im T)$. For $T$ in this class,

$\ber(T^n)\le(1+\sin^2\theta)^{n-1}\ber^n(T),\quad n\in\mathbb N.$

4. Geometric Characteristics of the Berezin Range

The Berezin range $\Ber(T)$ reflects the geometric distribution of the spectrum as measured against kernel functions. On the Dirichlet space $\mathcal D$:

• For finite-rank “block” operators $T(f)=\sum_{j=1}^n\langle f,g_j\rangle\,g_j$ with $g_j\in\mathcal D$, $\widetilde T(z)\in\mathbb R$ and $\Ber(T)$ is always a convex interval on the real axis.
• For finite-rank off-diagonal operators $T(f)=\sum\langle f,g_j\rangle\,h_j$ with real Taylor coefficients, the Berezin range is symmetric with respect to the real axis: $\overline{\widetilde T(\bar z)} = \widetilde T(z)$.
• For weighted shift operators $(Tz^n)=\beta_{n+1}z^{n+1}$,

$$\widetilde T(\lambda) = \frac{|\lambda|^2}{\ln(1/(1-|\lambda|^2))} \lambda \sum_{n=1}^\infty |\lambda|^{2n}\beta_{n+1}.$$

If $\beta_n\in\mathbb R$, $\Ber(T)$ is symmetric about the real axis; if $\beta_n\in i\mathbb R$, symmetry is about the imaginary axis. For $\beta_n=c/n$, $\widetilde T(\lambda)=c\lambda$, so $\Ber(T)$ is a closed disk centered at the origin of radius $|c|$.

5. Weak Power Bounds for Berezin Sectorial Operators

On certain spaces, specific power inequalities have been established. For $T\in\Pi^{\rm Ber,P}_\theta$ (the subclass characterized by geometric power-compatibility of the real and imaginary parts), the bound

$\ber(T^n) \le (1+\sin^2\theta)^{n-1}\,\ber^n(T), \quad n\in\mathbb N$

sharpens the familiar submultiplicative property and connects spectral parameters arising from the Berezin range with algebraic growth. For Toeplitz operators on weighted Bergman spaces with harmonic symbols,

$\ber(T_\phi^n) \le \ber^n(T_\phi)$

holds for all $n$.

6. Open Problems and Research Directions

Fundamental questions remain regarding the extension and application of Berezin sectoriality. Specifically, for composition-differentiation operators on the Dirichlet space $\mathcal D$, it is not yet established whether operators analogous to those in $H^2$—namely, Berezin–sectorial but not classically sectorial, or with sharply different sector indices via scalar shifts—can be constructed.

The analyticity of $\widetilde{D_\phi}(re^{i\theta})=f(r)e^{i\theta}$ suggests that, whenever $\lim_{r\to 1^-}f(r)$ exists, $\Ber(D_\phi)$ is a closed disk. Determining concrete instances where this disk's radius is strictly less than the numerical radius remains open. Further avenues include developing Berezin–sectorial theory in other reproducing-kernel contexts such as the Bergman, Fock, or de Branges–Rovnyak spaces and establishing precise bounds for the Berezin number in these broader settings (Mahapatra et al., 6 Jan 2026).