Email this article Orthogonality in nonseparable rearrangement-invariant spaces
- Authors: Astashkin S.V.1, Semenov E.M.2
-
Affiliations:
- Samara National Research University
- Voronezh State University
- Issue: Vol 212, No 11 (2021)
- Pages: 55-72
- Section: Articles
- URL: https://journals.rcsi.science/0368-8666/article/view/133473
- DOI: https://doi.org/10.4213/sm9543
- ID: 133473
Cite item
Open Access
Access granted
Subscription Access
Abstract
Let $E$ be a nonseparable rearrangement-invariant space and let $E_0$ be the closure of the space of bounded functions in $E$. Elements of $E$ orthogonal to $E_0$, that is, elements $x\in E$, $x\ne 0$, such that $\|x\|_{E} \le\|x+y\|_{E}$ for each $y\in E_0$, are investigated. The set of orthogonal elements $\mathcal{O}(E)$ is characterized in the case when $E$ is a Marcinkiewicz or an Orlicz space. If an Orlicz space $L_M$ is considered with the Luxemburg norm, then the set $L_M\setminus (L_M)_0$ is the algebraic sum of $\mathcal{O}(L_M)$ and $(L_M)_0$. Each nonseparable rearrangement-invariant space $E$ such that $\mathcal{O}(E)\ne\varnothing$ is shown to contain an asymptotically isometric copy of the space $l_\infty$. Bibliography: 17 titles.
About the authors
Sergei Vladimirovich Astashkin
Samara National Research University
Email: astash@ssau.ru
Doctor of physico-mathematical sciences, Professor
Evgenii Mikhailovich Semenov
Voronezh State University
Email: nadezhka_ssm@geophys.vsu.ru
Doctor of physico-mathematical sciences, Professor
References
- G. Birkhoff, “Orthogonality in linear metric spaces”, Duke Math. J., 1:2 (1935), 169–172
- B. D. Roberts, “On the geometry of abstract vector spaces”, Tôhoku Math. J., 39 (1934), 42–59
- R. C. James, “Orthogonality in normed linear spaces”, Duke Math. J., 12:2 (1945), 291–302
- E. W. Cheney, D. E. Wulbert, “The existence and unicity of best approximations”, Math. Scand., 24:1 (1969), 113–140
- I. Singer, Best approximation in normed linear spaces by elements of linear subspaces, Grundlehren Math. Wiss., 171, Publishing House of the Academy of the Socialist Republic of Romania, Bucharest; Springer-Verlag, New York–Berlin, 1970, 415 pp.
- П. А. Бородин, “Квазиортогональные множества и условия гильбертовости банахова пространства”, Матем. сб., 188:8 (1997), 63–74
- F. B. Saidi, “Characterisations of orthogonality in certain Banach spaces”, Bull. Austral. Math. Soc., 65:1 (2002), 93–104
- F. B. Saidi, “An extension of the notion of orthogonality to Banach spaces”, J. Math. Anal. Appl., 267:1 (2002), 29–47
- С. В. Асташкин, Е. М. Семeнов, “Об одном свойстве симметричных пространств, второе ассоциированное пространство к которым несепарабельно”, Матем. заметки, 107:1 (2020), 11–22
- С. В. Асташкин, Е. М. Семeнов, “Ортогональные элементы несепарабельных перестановочно-инвариантных пространств”, Докл. РАН. Мат. информ. проц. упр., 495:1 (2020), 5–7
- J. Lindenstrauss, L. Tzafriri, Classical Banach spaces, Ergeb. Math. Grenzgeb., II, Function spaces, Springer-Verlag, Berlin–New York, 1979, x+243 pp.
- С. Г. Крейн, Ю. И. Петунин, Е. М. Семенов, Интерполяция линейных операторов, Наука, М., 1978, 400 с.
- C. Bennett, R. Sharpley, Interpolation of operators, Pure Appl. Math., 129, Academic Press, Inc., Boston, MA, 1988, xiv+469 pp.
- М. А. Красносельский, Я. Б. Рутицкий, Выпуклые функции и пространства Орлича, Физматгиз, М., 1958, 271 с.
- M. M. Rao, Z. D. Ren, Theory of Orlicz spaces, Monogr. Textbooks Pure Appl. Math., 146, Marcel Dekker, Inc., New York, 1991, xii+449 pp.
- Е. В. Токарев, “О подпространствах некоторых симметричных пространств”, Теория функций, функциональный анализ и их приложения, 24, Вища школа, Харьков, 1975, 156–161
- P. N. Dowling, N. Randrianantoanina, “Asymptotically isometric copies of $ell^infty$ in Banach spaces and a theorem of Bessaga and Pelczynski”, Proc. Amer. Math. Soc., 128:11 (2000), 3391–3397
Supplementary files
