Quadratic forms in unitary operators

Let u1,…,un be unitary matrices on l2. Denote by the matrix A defined by A[(i, i′), (j, j′)] = Σk=1n uk(i, j)uk(i′, j′), acting as a bounded operator on . In other words, A is the sum of the Kronecker products of uk with their complex conjugates. We show the following sharp inequality: . As an application, we show that the natural representation ρ of U(N) (N 1), acting on L2 of the unit sphere in CN and restricted to mean zero functions, satisfies for any choice ω1,…,ωn in U(N) the lower bound . This extends a result due to Lubotzky, Phillips, and Sarnak, who proved this with SO(3) in the place of U(N).