On the characterization of Hankel-\(K\{M_p\}\) spaces in terms of the Zemanian differential operator

Abstract

For \(\mu \ge -\frac{1}{2}\), we show that membership in a space \(\mathcal {K}_\mu \) of type Hankel-\(K\{M_p\}\) can be characterized by separate boundedness conditions on a test function and on its \(T_{\mu , k}\)-derivatives, where, for every \(k \in \mathbb {N}\), \(T_{\mu , k}=N_{\mu +k-1} \ldots N_\mu \) is a suitable iterate of the Zemanian differential operator \(N_\mu =x^{\mu +\frac{1}{2}} D_x x^{-\mu -\frac{1}{2}}\), while \(T_{\mu , 0}\) corresponds to the identity operator. Besides yielding a new representation for the elements, the (weakly, weakly*, strongly) bounded subsets and the (weakly, weakly*, strongly) convergent sequences in the dual space \(\mathcal {K}_\mu ^{\prime }\), such a characterization ultimately proves that \(\mathcal {K}_\mu \) consists of all those functions in the Zemanian space \(\mathcal {H}_\mu \) whose product against every weight in the defining sequence \(\{M_p\}_{p=0}^\infty \) remains bounded.