Geodesic
The local ergodic theorem for groups of unitary operators and second order stationary processes
Sbornik. Mathematics, Tome 39 (1981) no. 2, pp. 227-242 
V. F. Gaposhkin. The local ergodic theorem for groups of unitary operators and second order stationary processes. Sbornik. Mathematics, Tome 39 (1981) no. 2, pp. 227-242. http://geodesic.mathdoc.fr/item/SM_1981_39_2_a4/
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     title = {The local ergodic theorem for groups of unitary operators and second order stationary processes},
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $(U_t)^\infty_{-\infty}$, be a strongly continuous unitary group in $L_2(X,S,\mu)$, where $\mu$ is a $\sigma$-finite measure. The local ergodic theorem is the relation \begin{equation} \lim_{t\to0}\frac1t\int^t_0(U_\tau f)(x)\,d\tau=f(x)\quad \text{a.\,e.} \end{equation} for $f\in L_2(X)$. It is shown that this relation is not satisfied for all $f\in L_2(X)$ and $\{U_t\}$. Necessary and sufficient conditions are obtained for the local ergodic theorem in terms of properties of the spectral measure $\{E(d\lambda)f\}$, where $\{E(d\lambda)\}$ is the resolution of the identity corresponding to the group $(U_t)$. In particular, (1) is satisfied if the integral $$ \int^\infty_{-\infty}[\log\log(\lambda^2+2)]^2\cdot\|E(d\lambda)f\|^2 $$ converges. Generalizations to multiparameter groups and homogeneous random fields are given. Bibliography: 10 titles.

[1] N. Danford, Dzh. Shvarts, Lineinye operatory, t. 1, IL, Moskva, 1962

[2] N. Wiener, “The ergodic theorem”, Duke Math. J., 5:1 (1939), 1–18 | DOI | MR | Zbl

[3] U. Krengel, “A local ergodic theorem”, Invent. math., 6 (1969), 329–333 | DOI | MR | Zbl

[4] T. R. Terrell, “Local ergodic theorems for $N$-parameter semigroups of operators”, Lecture Notes Math., 160 (1970), 53–63 | DOI | MR

[5] V. F. Gaposhkin, “Kriterii usilennogo zakona bolshikh chisel dlya klassov statsionarnykh protsessov i odnorodnykh sluchainykh polei”, Teoriya veroyatnostei, XXII:2 (1977), 295–319

[6] V. F. Gaposhkin, “Odna teorema o skhodimosti posledovatelnosti izmerimykh funktsii i ee primeneniya k posledovatelnostyam stokhasticheskikh integralov”, Matem. sb., 104(146) (1977), 3–21 | Zbl

[7] V. F. Gaposhkin, “Tochnye otsenki skorosti skhodimosti v usilennom zakone bolshikh chisel dlya klassov statsionarnykh v shirokom smysle posledovatelnostei i protsessov”, Uspekhi matem. nauk, XXXI:5(191) (1976), 233–234

[8] Y. Kubokawa, “Ergodic theorems for contraction semi-groups”, J. Math. Soc. Japan, 27:2 (1975), 184–193 | DOI | MR | Zbl

[9] Y. Kubokawa, “A general ergodic theorem”, Proc. Japan Acad. Sci., 48 (1972), 461–465 | DOI | MR | Zbl

[10] R. Sato, “On a local ergodic theorem”, Studia Math., 58 (1976), 1–5 | MR | Zbl