Some remarks on the asymptotic behaviour of the iterates of a bounded linear operator

Anatoliĭ Antonevich; Jürgen Appell; Petr Zabreĭko

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We prove that an Orlicz space equipped with the Luxemburg norm has uniformly normal structure if and only if it is reflexive.

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In [2] and [3] upper and lower estimates and asymptotic results were obtained for the approximation numbers of the operator $T : L^{p} (ℝ^{+}) \to L^{p} (ℝ^{+})$ defined by $(T f) (x) ≔ v (x) ʃ_{0}^{\infty} u (t) f (t) d t$ when 1 < p < ∞. Analogous results are given in this paper for the cases p = 1,∞ not included in [2] and [3].

Jung constants of Orlicz function spaces.

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Estimation of the Jung constants of Orlicz function spaces equipped with either Luxemburg norm or Orlicz norm is given. The exact values of the Jung constants of a class of reflexive Orlicz function spaces have been found by using a new quantitative index of N-functions.

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