Berezin Sectorial Operators Overview

• Berezin sectorial operators are bounded linear operators on reproducing kernel Hilbert spaces defined via the Berezin transform, emphasizing a refined Berezin range.
• They generalize classical sectorial operators by replacing the numerical range with a tighter geometric support, which leads to sharper operator inequalities.
• The framework yields new comparison results and improved spectral estimates, with applications in spaces such as Hardy-Hilbert and Dirichlet.

Berezin sectorial operators constitute a novel class of bounded linear operators on reproducing kernel Hilbert spaces, generalizing classical sectorial operators by means of the Berezin transform. Rather than controlling the numerical range, Berezin sectorial operators are characterized via the geometry of their Berezin range—a typically smaller and subtler region in the complex plane—which leads to distinct operator-theoretic and geometric phenomena. This approach yields new comparison results, sharp inequalities for the Berezin number, and offers novel perspectives on structure and index reduction, particularly in contexts such as the Hardy-Hilbert and Dirichlet spaces (Mahapatra et al., 6 Jan 2026).

1. Formal Definition and Foundational Framework

Let $\mathcal{K}$ be a reproducing-kernel Hilbert space over a set $\Omega$, with normalized kernels $\{\hat{k}_\lambda: \lambda \in \Omega\}$. For any $T \in \mathscr{B}(\mathcal{K})$ the Berezin transform of $T$ is

$$\widetilde{T}(\lambda) = \langle T \hat{k}_\lambda, \hat{k}_\lambda \rangle, \qquad Ber(T) = \{\widetilde{T}(\lambda): \lambda \in \Omega\}, \qquad ber(T) = \sup_{\lambda \in \Omega} |\widetilde{T}(\lambda)|.$$

Fix an angle $\theta \in [0, \tfrac{\pi}{2})$ and the sector $$S_\theta = \{z \in \mathbb{C}: |\arg z| \leq \theta\}$$. $T$ is Berezin sectorial of semi-angle $\theta$, denoted $T \in \Pi^{Ber}_{\theta}$, if

$Ber(T) \subseteq S_\theta.$

Equivalently, for all $|\phi|<\theta$, $\widetilde{e^{-i\phi} T}(\lambda)$ lies in the right half-plane for all $\lambda$.

2. Relationship with Classical Sectorial Operators

Classically, an operator $T$ on a Hilbert space is sectorial of angle $\theta$ if its numerical range

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

satisfies $W(T) \subseteq S_\theta$. Berezin sectoriality is strictly weaker:

• In $H^2(\mathbb{D})$, for $D_\phi(f)=f'\circ\phi$ with $\phi(z)=\rho z$, $0<\rho<1$, one finds

$$Ber(D_\phi) = \overline{B(0, r_1(\rho))}, \qquad W(D_\phi) = \overline{B(0, w(D_\phi))}$$

where $r_1(\rho)$ and $w(D_\phi)$ are given by semiexplicit formulas and inequalities. Numerically, $r_1(\rho)<\tfrac{1}{2}\|D_\phi\| \leq w(D_\phi)$, resulting in $Ber(D_\phi) \subsetneq W(D_\phi)$.

• By adding a real shift $\alpha I$, one constructs operators that are Berezin sectorial but not sectorial in the classical sense. Even when classical sectoriality holds, the Berezin sectorial index (the minimal $\theta$) can be strictly smaller due to the smaller geometric support: the Berezin range may fit inside a cone strictly contained in the one containing the numerical range.

A plausible implication is that operator inequalities or spectral inclusions based on sectoriality can be sharpened when the Berezin transform is considered instead of the numerical range.

3. Berezin Number Inequalities: Angle and Product Bounds

The Berezin number $ber(T)$, central to analysis, admits a variety of inequalities for Berezin sectorial operators $T\in\Pi^{Ber}_\theta$, $0<\theta<\frac{\pi}{2}$.

• Angle lemma: $\sin\theta \cdot ber(T) \geq ber(\Im T)$ This provides a lower bound for the imaginary part in terms of the Berezin number and the sectorial angle.
• Lower bound for $ber(T)$: $$ber(T) \geq \frac{\csc\theta}{2} \, ber(\Re T \pm \Im T) + \frac{\csc\theta}{2} \left(ber(\Im T)-ber(\Re T)\right)$$
• Product estimate: If $T^* S \in \Pi^{Ber}_\theta$, for any $\alpha>0$,

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

with

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

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

• Norm-type bounds: For invertible $S$ with $S^* T \in \Pi^{Ber}_\theta$,

$$\||T|^2\|_{ber}^{1/2} \leq \|S^{-1}\| \left(\sin \theta\, ber(S^* T) + \frac{1}{2} \||T-iS|^2\|_{ber}\right)$$

$$\||T|^2\|_{ber} \leq \frac{1}{4}(1+\sin\theta)^2\, ber^2(T) + \frac{1}{2} \inf_{t\in\mathbb{R}}\left(\|T-tI\|_{ber}^2 + \|T-itI\|_{ber}^2 \right)$$

• Weak power-type inequalities: For $T$ in the subclass

$$\Pi^{Ber,P}_\theta = \{ T\in\Pi^{Ber}_\theta : ber(\Re T^n) \leq ber^n(\Re T),\, ber(\Im T^n)\leq ber^n(\Im T)\; \forall n\in\mathbb{N} \}$$

$$ber(T^n) \leq (1+\sin^2\theta)^{n-1} \, ber^n(T), \qquad n=1, 2, \ldots$$

Notably, for Toeplitz operators $T_\varphi$ on weighted Bergman spaces with harmonic symbol, the power-inequality $ber(T_\varphi^n) \leq ber^n(T_\varphi)$ is exact.

4. Geometry of the Berezin Range

The structure of $Ber(T)$ strongly influences both sectoriality and inequality sharpness.

• Finite-rank operators on the Dirichlet space $\mathcal{D}$: For $T(f) = \sum_{j=1}^n \langle f, g_j \rangle g_j$, $g_j \in \mathcal{D}$,

$$\widetilde{T}(\lambda) = |\lambda|^2\, \frac{1}{\ln\left(\tfrac{1}{1-|\lambda|^2}\right)} \sum_{j=1}^n |g_j(\lambda)|^2 \in \mathbb{R}$$

Thus $Ber(T)\subset \mathbb{R}$, an interval, and hence convex in $\mathbb{C}$.

• General symmetry: For $T(f)=\sum \langle f, g_j \rangle h_j$ where $g_j, h_j$ have real Taylor coefficients, $Ber(T)$ is symmetric about the real axis.
• Weighted shifts on $\mathcal{D}$: For $T(\sum a_n z^n) = \sum a_n \beta_{n+1} z^{n+1}$

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

Symmetry properties vary with the coefficients: real $\beta_n$ yield symmetry about the real axis, purely imaginary $\beta_n$ about the imaginary axis. For $\beta_n = c/n$ ($c\in\mathbb{D}$), $Ber(T)$ is a disk of radius $|c|$.

This suggests that Berezin sectoriality is sensitive to symmetry conditions and spatial supports of the transform.

5. Applications and Operator Examples

Concrete construction of Berezin sectorial operators is illustrated on the Hardy-Hilbert space $H^2(\mathbb{D})$ via composition-differentiation operators

$$D_\phi(f) = f' \circ \phi, \qquad \phi(z) = \rho z, \; 0<\rho<1$$

with explicit calculations of Berezin and numerical ranges showing the existence of operators that are Berezin sectorial but not classical sectorial.

A plausible implication is that Berezin sectoriality permits sharp control for operator families where traditional sectorial tools may fail, especially for operator inequalities and spectral localization.

6. Open Problems and Future Research

The systematic construction of Berezin sectorial composition-differentiation operators on the Dirichlet space remains unresolved. Critical open questions include:

• For an analytic self-map $\phi$ of $\mathbb{D}$, finding conditions ensuring $D_\phi$ is Berezin sectorial of angle $\theta$ but not classically sectorial.
• Geometric criteria on the Berezin transform that guarantee a strictly smaller Berezin index than the numerical range index for composition-differentiation operators.

These open questions underscore the intricate relationships between Berezin transform geometry, sectoriality indices, and associated operator inequalities, signaling directions for deeper study of functional calculus, spectral theory, and operator geometry (Mahapatra et al., 6 Jan 2026).