2017/12/13 by Karl‐Mikael Perfekt, Karl-Mikael Perfekt, Perfekt, Karl-Mikael
Mathematics · #46B28 (Primary) 47B35 (Secondary) #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics #math.FA #msc:46B28 #msc:47B35
paper · pdf · doi:10.48550/arxiv.1712.04894
11 pages
arxiv created 2017/12/13 · openalex publication_date 2017/12/13 · arxiv updated 2017/12/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A multiplicative Hankel operator is an operator with matrix representation M(α) = \α(nm)\n,m=1^∞, where α is the generating sequence of M(α). Let M and M0 denote the spaces of bounded and compact multiplicative Hankel operators, respectively. In this note it is shown that the distance from an operator M(α) ∈ M to the compact operators is minimized by a nonunique compact multiplicative Hankel operator N(β) ∈ M0, ‖M(α) - N(β)‖B(ℓ2(ℕ)) = inf \‖M(α) - K ‖B(ℓ2(ℕ)) : K \colon ℓ2(ℕ) → ℓ2(ℕ) \textrm compact \. Intimately connected with this result, it is then proven that the bidual of M0 is isometrically isomorphic to M, M0∗ ∗ ≃ M. It follows that M0 is an M-ideal in M. The dual space M0^∗ 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 H2(\mathbbDd) of a finite polydisk.