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

Authors

Azizkhon Azizov, National University of UzbekistanFollow

Abstract

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 C_E be the symmetric ideal of compact operators in H generated by the fully symmetric sequence space E ⊂ c₀. If T_u: 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 ∈ C_E the net A_t(y) = 1/t^d ∫_[0,t]^d T _u(y) du, t>0, converges to some ŷ ∈ C_E with respect to the norm ||.||_∞, as t → ∞; moreover, if E is separable \ and E ≠ l₁ (as a sets), then A_t(y) converges to ŷ with respect to the norm ||.||_CE.

First Page

44

Last Page

53

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Recommended Citation

Azizov, Azizkhon (2021) "Ergodic theorems for d-dimensional flows in ideals of compact operators," Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences: Vol. 4: Iss. 1, Article 4.
DOI: https://doi.org/10.56017/2181-1318.1150