On the convexity of the Berezin range of composition operators and related questions

Abstract

The Berezin range of a bounded operator $T$ acting on a reproducing kernel Hilbert space $\mathcal{H}$ is the set $B(T)$ := $\{\langle T\hat{k}_{x},\hat{k}_{x} \rangle_{\mathcal{H}} : x \in X\}$, where $\hat{k}_{x}$ is the normalized reproducing kernel for $\mathcal{H}$ at $x \in X$. In general, the Berezin range of an operator is not convex. Primarily, we focus on characterizing the convexity of the Berezin range for a class of composition operators acting on the Fock space on $\mathbb{C}$ and the Dirichlet space of the unit disc $\mathbb{D}$. We prove an analogue of the elliptic range theorem for the unitarily equivalent Berezin range of an operator on a two-dimensional reproducing kernel Hilbert space and characterize the convexity of the unitarily equivalent Berezin range for a bounded operator $T$ on a reproducing kernel Hilbert space $\mathcal{H}$.

Figures (5)

• Figure 1: $B(C_\phi)$ on $F^2(\mathbb{C})$ for $\zeta=0.5e^{i\frac{\pi}{3}}$(apparently not convex).
• Figure 2: $B(C_\phi)$ on $F^2(\mathbb{C})$ for $a=e^{i\frac{\pi}{12}}$ and $b=0$(apparently not convex).
• Figure 3: $\text{Ber}(C_\phi)$ on $F^2(\mathbb{C})$ for $\zeta=0.5$ and $a=1$(left, apparently not convex) and for $\zeta=0.5$ and $a=10$(right, apparently convex).
• Figure 4: $B(C_\phi)$ on $\mathcal{D}$ for $\zeta=-i$ (apparently not convex).
• Figure 5: $\text{Ber}(C_{\phi_\alpha})$ on $\mathcal{D}$ for $\alpha=\frac{1}{2}e^{(i\frac{\pi}{3})}$ (apparently not convex).

Theorems & Definitions (22)

• Theorem 2.1
• Theorem 2.2
• proof
• Corollary 2.3
• Remark 2.4
• Theorem 3.1: el2014primer, Section 1.4
• Theorem 3.2
• proof
• Proposition 3.3
• proof