Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences


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.

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