Welcome to Trieu Le's homepage

We explore properties of the Berezin range $B(C_\phi)$ of a composition operator $C_\phi$ on the Hardy space $H^2$, including constraints on its location in the complex plane $\mathbb{C}$, topological properties, shape, symmetry, and convexity. For example, we show that$B(C_\phi) \subseteq \{z\in \mathbb{C}: \Re(z) > 0 \ \text{and}\ |z| \le 1 + |\phi(0)|\},$ that $B(C_\phi)$ is closed if and only if $C_\phi$ is the identity operator, and that $B(C_\phi)$ contains all its boundary points except for $0$ if $C_\phi$ is compact. Addressing questions raised by Cowen and Felder, we show that $C_\phi$ can be a (nondegenerate) disk and characterize when $B(C_\phi)$ is convex if $\phi$ is linear.

We study compactness of product of Toeplitz operators with symbols continuous on the closure of the polydisc in terms of behavior of the symbols on the boundary. For certain classes of symbols $f$ and $g$, we show that $T_fT_g$ is compact if and only if $fg$ vanishes on the boundary. We provide examples to show that for more general symbols, the vanishing of $fg$ on the whole polydisc might not imply the compactness of $T_fT_g$. On the other hand, the reverse direction is closely related to the zero product problem for Toeplitz operators on the unit disc, which is still open.