 

KarlMikael Perfekt
On the spaces of bounded and compact multiplicative Hankel operators
view print


Published: 
July 29, 2019. 
Keywords: 
essential norm, Hankel operator, bidual, Mideal, weak product space. 
Subject: 
46B28, 47B35. 


Abstract
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 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(α) ∈ M to the compact operators is minimized by a nonunique compact multiplicative Hankel operator N(β) ∈ M_{0}.
Intimately connected with this result, it is then proven that the bidual of M_{0} is isometrically isomorphic to M, M_{0}** ≅ M. It follows that M_{0} is an Mideal in M. The dual space M_{0}* 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}(D^{d}) of a finite polydisk. 

Acknowledgements
N/A.


Author information
KarlMikael Perfekt:
Department of Mathematics and Statistics
University of Reading
Reading RG6 6AX, United Kingdom
k.perfekt@reading.ac.uk

