 

Rupert H. Levene
Norms of idempotent Schur multipliers view print


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


Abstract
Let D be a masa in B(H) where H is a separable Hilbert
space. We find real
numbers η_{0}<η_{1}<η_{2}<...<η_{6} so that for every
bounded, normal Dbimodule map Φ on B(H),
either ∥Φ∥>η_{6} or
∥Φ∥=η_{k}
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 η_{k} 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

