Asymptotic and Partial Asymptotic Hankel Operators on $H^2\left(D^n\right)$)

In this paper, we generalize the concept of asymptotic Hankel operators on $H^2\left(D\right)$ to the Hardy space $H^2\left(D^n\right)$ (over polydisk) in terms of asymptotic Hankel and partial asymptotic Hankel operators and investigate some properties in case of its weak and strong convergence. Meanwhile, we introduce i^th-partial Hankel operators on $H^2\left(D^n\right)$ and obtain a characterization of its compactness for n > 1. Our main results include the containment of Toeplitz algebra in the collection of all strong partial asymptotic Hankel operators on $H^2\left(D^n\right)$. It is also shown that a Toeplitz operator with symbol ϕ is asymptotic Hankel if and only if ϕ is holomorphic function in $L^{\infty }\left(T^n\right)$.