On the spaces of bounded and compact multiplicative Hankel operators

A multiplicative Hankel operator is an operator with matrix representation $M(\alpha )=\{\alpha (nm){\}}_{n,m=1}^{\mathrm{\infty }}$, 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}$$ to the compact operators is minimized by a nonunique compact multiplicative Hankel operator $N(\beta )\in {\mathcal{M}}_0$, $$\mathrm{\parallel }M(\alpha )-N(\beta ){\mathrm{\parallel }}_{\mathcal{B}({\mathrm{\ell }}^2(\mathbb{N}))}=\inf \{\mathrm{\parallel }M(\alpha )-K{\mathrm{\parallel }}_{\mathcal{B}({\mathrm{\ell }}^2(\mathbb{N}))}:K\text{\hspace{0.17em}⁣}:{\mathrm{\ell }}^2(\mathbb{N})\to {\mathrm{\ell }}^2(\mathbb{N})\text{ compact}\}\mathrm{.}$$ 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.