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Statistical ergodic theorem in symmetric spaces for infinite measures
A. S. Vekslera, V. I. Chilinb a Institute of Mathematics of the Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan
b National University of Uzbekistan named after M. Ulugbek, Tashkent, Uzbekistan
Abstract:
Let (Ω,μ) be a measurable space with σ-finite continuous measure, μ(Ω)=∞. A linear operator T:L1(Ω)+L∞(Ω)→L1(Ω)+L∞(Ω) is called the 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)=1tt∫0Ts(f)ds∈L1(Ω), f∈L1(Ω) has a unique extension to the Dunford–Schwartz operator, which is also denoted by At, t>0. It is proved that in the completely symmetric space E(Ω)⊈ of measurable functions on (\Omega, \mu) the means A_t converge strongly as t\to +\infty for each strongly continuous in L_1(\Omega) semigroup \{T_t\}_{t\geqslant 0} of Dunford–Schwartz operators if and only if the norm \|\cdot\|_{E(\Omega)} is order continuous.
Citation:
A. S. Veksler, V. I. Chilin, “Statistical ergodic theorem in symmetric spaces for infinite measures”, Science — Technology — Education — Mathematics — Medicine, CMFD, 67, no. 4, PFUR, M., 2021, 654–667
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https://www.mathnet.ru/eng/cmfd441 https://www.mathnet.ru/eng/cmfd/v67/i4/p654
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Abstract page: | 155 | Full-text PDF : | 73 | References: | 38 |
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