  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.
