New York Journal of Mathematics
Volume 20 (2014) 325-352


Rupert H. Levene

Norms of idempotent Schur multipliers

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Published: April 7, 2014
Keywords: idempotent Schur multiplier, normal masa bimodule map, Hadamard product, norm, bipartite graph
Subject: 47A30, 15A60, 05C50

Let D be a masa in B(H) where H is a separable Hilbert space. We find real numbers η012<...<η6 so that for every bounded, normal D-bimodule map Φ on B(H), either ∥Φ∥>η6 or ∥Φ∥=ηk for some k ∈ {0,1,2,3,4,5,6}. When D is totally atomic, these maps are the idempotent Schur multipliers and we characterise those with norm ηk for 0 ≦ k ≦ 6. We also show that the Schur idempotents which keep only the diagonal and superdiagonal of an n × n matrix, or of an n × (n+1) matrix, both have norm (2/(n+1))cot(π/(2(n+1))), and we consider the average norm of a random idempotent Schur multiplier as a function of dimension. Many of our arguments are framed in the combinatorial language of bipartite graphs.

Author information

School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland