On the spaces of bounded and compact multiplicative Hankel operators

摘要

A multiplicative Hankel operator is an operator with matrix representation M() = {(nm)}n,m=1, where is the generating sequence of M(). 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() M to the compact operators is minimized by a nonunique compact multiplicative Hankel operator N() 8 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.