New York Journal of Mathematics
Volume 25 (2019), 589-602


Karl-Mikael Perfekt

On the spaces of bounded and compact multiplicative Hankel operators

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

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. 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(Dd) of a finite polydisk.



Author information

Karl-Mikael Perfekt:
Department of Mathematics and Statistics
University of Reading
Reading RG6 6AX, United Kingdom