A pointwise ergodic theorem for contractions in \(\mathbb L_p({\mathcal H})\)-spaces (Q2498302)

Let \(\mathcal H\) be a complex separable Hilbert space and \({\L}_p (Z,{\mathcal F},\mu ;{\mathcal H} )\) denote the space of \(\mathcal H\)-valued measurable functions \(\xi\) over a finite measure space \((Z,{\mathcal F},\mu )\) such that the norm \(| | \xi | | _P \) is finite. The main result of the present paper reads as follows. Theorem. Let \(T\) be a contraction on \({\mathbb L}_P (Z,{\mathcal F},\mu ;{\mathcal H} )\), \(1<p<\infty\), \(p\neq 2\). Suppose that for each finite partition \(G=(E_1 ,\dots, E_n )\) of \(Z\) (with \(\mu (E_j )=m_j\)) and the corresponding conditional expectation \({\mathcal E}_G\), the contraction \({\mathcal E}^G T\) on \({\mathbb L}_p (Z,S,\mu,{\mathcal H})\sim {\mathbb L}_P (m_1,\dots,m_n ,{\mathcal H})\) is symmetric and contractively majorized. Then the Cesàro means \(\frac{1}{n} \sum_{i=0}^{n-1} (T^k f)(z), n={\overline {1,\infty}}\), converge strongly \(\mu\)-almost everywhere in \(\mathcal H\). This result is closely related to the ergodic theorem of \textit{M.\,A.\,Akcoglu} [Can.\ J.\ Math.\ 27, 1075--1082 (1975; Zbl 0326.47005)] for positive contractions in the classical \(L_p\)-spaces.