On the dominated ergodic theorem in $L_{2}$ space

Abstract:

Let $T$ be a contraction on ${L_2}$ of a $\sigma$-finite measure space, ${A_n}(T)$ the operator $(1/n)({T^0} + \cdots + {T^n}),S(T)f$ the function ${\sup _n}|{A_n}(T)f|$. Theorem 1. Assume that, whatever be the measure space, $S(U)f \in {L_2}$ for each unitary operator $U$ on ${L_2}$ and each function $f \in {L_2}$. Then there exists a universal constant $K$ such that $||S(T)f|| \leqq K||f||$ for each contraction $T$ on ${L_2}$ and each $f \in {L_2}$. Theorem 2. Let $T$ be a contraction on ${L_2}$ and let $U$ be a unitary dilation of $T$ acting on a Hilbert space $H$ containing ${L_2}$. If all expressions of the form $\Sigma _{n = 1}^\infty {{P_n}{A_n}(U)}$, where ${P_n}$ are mutually orthogonal projections, are bounded operators on $H$, then for each $f \in {L_2}, S(T)f \in {L_2}$ and ${A_n}(T)f$ converges a.e.