Quadratic forms in unitary operators Academic Article uri icon

abstract

  • Let u1, . . . , un be unitary matrices on l2. Denote by nk = 1 uk k the matrix A defined by A[(i, i), (j, j)] = nk = 1 uk (i, j)uk (i, j), acting as a bounded operator on l2( X ). In other words, A is the sum of the Kronecker products of uk with their complex conjugates. We show the following sharp inequality: nk = 1 uk k 2n - 1 . As an application, we show that the natural representation of U(N) (N 1), acting on L2 of the unit sphere in CN and restricted to mean zero functions, satisfies for any choice 1, . . . , n in U(N) the lower bound n1 (k) 2n - 1. This extends a result due to Lubotzky, Phillips, and Sarnak, who proved this with SO(3) in the place of U(N). 1997 Elsevier Science Inc.

published proceedings

  • LINEAR ALGEBRA AND ITS APPLICATIONS

author list (cited authors)

  • Pisier, G.

citation count

  • 3

complete list of authors

  • Pisier, G

publication date

  • December 1997