Ergodic theorems for d-dimensional flows in ideals of compact operators

Let H be an infinite-dimensional complex Hilbert space, let (B(H), ‖ · ‖∞) be the C?-algebra of all bounded linear operators acting in H, and let CE be the symmetric ideal of compact operators in H generated by the fully symmetric sequence space E ⊂ c0. If Tu : B(H) → B(H), u = (u1, . . . , ud) ∈ R+, is a semigroup of positive Dunford-Schwartz operators, which is strongly continuous on C1, then the following versions of individual and mean ergodic theorems are true: For each x ∈ CE the net At(x) = 1 td ∫ [0,t]d Tu(x)du, t > 0, converges to some x̂ ∈ CE with respect to the norm ‖ · ‖∞, as t → ∞; moreover, if E is separable and E 6= l1 (as a sets), then lim t→∞ ‖At(x)− x̂‖CE = 0.