Abstract
Let (Ω, μ) be a measurable space with a σ-finite continuous measure, μ(Ω) = ∞. A linear operator T : L1(Ω) + L∞(Ω) → L1(Ω) + L∞(Ω) is called a Dunford–Schwartz operator if ∥T(f)∥1 ≤ ∥f∥1 (respectively, ∥T(f)∥∞ ≤ ∥f∥∞) for all f ∈ L1(Ω) (respectively, f ∈ L∞(Ω)). If {Tt}t≥0 is a strongly continuous in L1(Ω) semigroup of Dunford–Schwartz operators, then each operator At(f) = \(\frac{1}{t}\underset{0}{\overset{t}{\int }}{T}_{s}\left(f\right)ds\in {L}_{1}\left(\Omega \right)\), f ∈ L1(Ω) has a unique extension to a Dunford–Schwartz operator, which is also denoted by At, t > 0. It is proved that in a completely symmetric space E(Ω) ⊈ L1 of measurable functions on (Ω, μ) the means At converge strongly as t → +∞ for each strongly continuous in L1(Ω) semigroup {Tt}t≥0 of Dunford–Schwartz operators if and only if the norm ∥·∥E(Ω) is order continuous.
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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 67, No. 4, Science — Technology — Education — Mathematics — Medicine, 2022.
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Veksler, A.S., Chilin, V.I. Statistical Ergodic Theorem in Symmetric Spaces for Infinite Measures. J Math Sci 278, 426–438 (2024). https://doi.org/10.1007/s10958-024-06931-6
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DOI: https://doi.org/10.1007/s10958-024-06931-6