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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.

We focus on two problems relating to the question of when the product of two posinormal operators is posinormal, giving (1) necessary conditions and sufficient conditions for posinormal operators to have closed range, and (2) sufficient conditions for the product of commuting closed-range posinormal operators to be posinormal with closed range. We also discuss the relationship between posinormal operators and EP operators (as well as hypo-EP operators), concluding with a new proof of the Hartwig-Katz Theorem, which characterizes when the product of posinormal operators on $\mathbb{C}^n$ is posinormal.

We find sufficient conditions for a self-map of the unit ball to converge uniformly under iteration to a fixed point or idempotent on the entire ball. Using these tools, we establish spectral containments for weighted composition operators on Hardy and Bergman spaces of the ball. When the compositional symbol is in the Schur-Agler class, we establish the spectral radii of these weighted composition operators.