On the spaces of bounded and compact multiplicative Hankel operators

Abstract

A multiplicative Hankel operator is an operator with matrix representation $M(\alpha) = \{\alpha(nm)\}_{n,m=1}^\infty$, where $\alpha$ is the generating sequence of $M(\alpha)$. Let $\mathcal{M}$ and $\mathcal{M}_0$ denote the spaces of bounded and compact multiplicative Hankel operators, respectively. In this note it is shown that the distance from an operator $M(\alpha) \in \mathcal{M}$ to the compact operators is minimized by a nonunique compact multiplicative Hankel operator $N(\beta) \in \mathcal{M}_0$, $$\|M(\alpha) - N(\beta)\|_{\mathcal{B}(\ell^2(\mathbb{N}))} = \inf \left \{\|M(\alpha) - K \|_{\mathcal{B}(\ell^2(\mathbb{N}))} \, : \, K \colon \ell^2(\mathbb{N}) \to \ell^2(\mathbb{N}) \textrm{ compact} \right\}.$$ Intimately connected with this result, it is then proven that the bidual of $\mathcal{M}_0$ is isometrically isomorphic to $\mathcal{M}$, $\mathcal{M}_0^{\ast \ast} \simeq \mathcal{M}$. It follows that $\mathcal{M}_0$ is an M-ideal in $\mathcal{M}$. The dual space $\mathcal{M}_0^\ast$ is isometrically isomorphic to a projective tensor product with respect to Dirichlet convolution. The stated results are also valid for small Hankel operators on the Hardy space $H^2(\mathbb{D}^d)$ of a finite polydisk.