Quadratic forms in unitary operators

Abstract

Let $u_1,\dots ,u_n$ be unitary operators on a Hilbert space $H$. We study the norm $$\left\|\sum^{i=n}_{i=1} u_i \otimes \bar u_i\right\|\leqno (1)$$ of the operator $\sum u_i\otimes {\bar{u}}_i$ acting on the Hilbertian tensor product $H{\otimes }_2\overline{H}$. The main result of this note is Theorem 1. For any $n$-tuple $u_1,\dots ,u_n$ of unitary operators in $B(H)$, we have $$2\sqrt{n-1} \le \left\|\sum^n_1 u_i \otimes \bar u_i\right\|.\leqno (6)$$ In other words, the right side of (6) is minimal exactly when $u_i=\lambda (g_i)$.