On the spaces of bounded and compact multiplicative Hankel operators

TL;DR

This paper characterizes the structure of bounded and compact multiplicative Hankel operators, showing the approximation properties, duality relations, and M-ideal structure, with extensions to multivariable Hardy spaces.

Contribution

It proves the minimal distance approximation by compact operators, establishes the bidual isomorphism, and describes the dual space structure of multiplicative Hankel operators.

Findings

The distance from a bounded multiplicative Hankel operator to the compact operators is minimized by a nonunique compact Hankel operator.

The bidual of the space of compact multiplicative Hankel operators is isometrically isomorphic to the space of bounded ones.

The dual space of compact multiplicative Hankel operators is isometrically isomorphic to a projective tensor product with Dirichlet convolution.

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.