Dominated estimates in Hilbert space

Abstract:

Let $U$ be a unitary operator on a Hilbert space $H$, and let ${A_n}(U),n = 1,2, \ldots$, be the Cesàaro means of $U$. It is shown that $\Sigma _{n = 1}^\infty {P_n}{A_n}(U)$ is bounded for every sequence of mutually orthogonal projections ${P_n},n = 1,2, \ldots$, if and only if $1$ is not a limit point of the spectrum of $U$. The proof is obtained by adapting ideas of Menchoff and Burkholder to show that for any orthonormal sequence ${f_n},n = 0, \pm 1, \pm 2, \ldots$, in $H$, there is an orthonormal sequence ${g_n},n = 1,2, \ldots$, such that \[ \sum \limits _{k = 1}^n {|({f_1} + {f_2} + \cdots + {f_k},{g_k}){|^2} \geqslant \frac {1} {{36}}n{{(\log n)}^2}.} \]