Illinois Journal of Mathematics

Norms of Schur multipliers

Kenneth R. Davidson and Allan P. Donsig

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A subset $\mathcal{P}$ of $\mathbb{N}^2$ is called Schur bounded if every infinite matrix with bounded scalar entries which is zero off of $\mathcal{P}$ yields a bounded Schur multiplier on $\mathcal{B}(\mathcal{H})$. Such sets are characterized as being the union of a subset with at most $k$ entries in each row with another that has at most $k$ entries in each column, for some finite $k$. If $k$ is optimal, there is a Schur multiplier supported on the pattern with norm $O(\sqrt k)$, which is sharp up to a constant. The same characterization also holds for operator-valued Schur multipliers in the cb-norm, i.e., every infinite matrix with bounded \emph{operator} entries which is zero off of $\mathcal{P}$ yields a completely bounded Schur multiplier. This result can be deduced from a theorem of Varopoulos on the projective tensor product of two copies of $l^\infty$. Our techniques give a new, more elementary proof of his result. We also consider the Schur multipliers for certain matrices which have a large symmetry group. In these examples, we are able to compute the Schur multiplier norm exactly. This is carried out in detail for a few examples including the Kneser graphs.

Article information

Illinois J. Math., Volume 51, Number 3 (2007), 743-766.

First available in Project Euclid: 13 November 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47Lxx: Linear spaces and algebras of operators [See also 46Lxx]
Secondary: 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05] 47Axx: General theory of linear operators


Davidson, Kenneth R.; Donsig, Allan P. Norms of Schur multipliers. Illinois J. Math. 51 (2007), no. 3, 743--766. doi:10.1215/ijm/1258131101.

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