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=(u_1,...,u_d)∈ R+d, 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 y ∈ CE the net At(y) = 1/td ∫[0,t]d T u(y) du, t>0, converges to some ŷ ∈ CE with respect to the norm ||.||∞, as t → ∞; moreover, if E is separable \ and E ≠ l1 (as a sets), then At(y) converges to ŷ with respect to the norm ||.||CE.

Hali tarjima qilinmagan