Orthogonality in nonseparable rearrangement-invariant spaces

S. V. Astashkin   ; E. M. Semenov

Sbornik. Mathematics, Tome 212 (2021) no. 11, pp. 1553-1570

Résumé

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.

Détail

Keywords: rearrangement-invariant space, nonseparable Banach space, Orlicz space
Mots-clés : Marcinkiewicz space, orthogonal element.

Bibliographie

[1] G. Birkhoff, “Orthogonality in linear metric spaces”, Duke Math. J., 1:2 (1935), 169–172 | DOI | MR | Zbl

[2] B. D. Roberts, “On the geometry of abstract vector spaces”, Tôhoku Math. J., 39 (1934), 42–59 | Zbl

[3] R. C. James, “Orthogonality in normed linear spaces”, Duke Math. J., 12:2 (1945), 291–302 | DOI | MR | Zbl

[4] E. W. Cheney, D. E. Wulbert, “The existence and unicity of best approximations”, Math. Scand., 24:1 (1969), 113–140 | DOI | MR | Zbl

[5] 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. | DOI | MR | Zbl

[6] P. A. Borodin, “Quasiorthogonal sets and conditions for a Banach space to be a Hilbert space”, Sb. Math., 188:8 (1997), 1171–1182 | DOI | DOI | MR | Zbl

[7] F. B. Saidi, “Characterisations of orthogonality in certain Banach spaces”, Bull. Austral. Math. Soc., 65:1 (2002), 93–104 | DOI | MR | Zbl

[8] F. B. Saidi, “An extension of the notion of orthogonality to Banach spaces”, J. Math. Anal. Appl., 267:1 (2002), 29–47 | DOI | MR | Zbl

[9] S. V. Astashkin, E. M. Semenov, “On a property of rearrangement invariant spaces whose second Köthe dual is nonseparable”, Math. Notes, 107:1 (2020), 10–19 | DOI | DOI | MR | Zbl

[10] S. V. Astashkin, E. M. Semenov, “Orthogonal elements in nonseparable rearrangement invariant spaces”, Dokl. Math., 102:3 (2020), 449–450 | DOI | DOI

[11] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces, Ergeb. Math. Grenzgeb., II, Function spaces, Springer-Verlag, Berlin–New York, 1979, x+243 pp. | MR | Zbl

[12] S. G. Kreĭn, Yu. I. Petunin, E. M. Semenov, Interpolation of linear operators, Transl. Math. Monogr., 54, Amer. Math. Soc., Providence, RI, 1982, xii+375 pp. | MR | MR | Zbl | Zbl

[13] C. Bennett, R. Sharpley, Interpolation of operators, Pure Appl. Math., 129, Academic Press, Inc., Boston, MA, 1988, xiv+469 pp. | MR | Zbl

[14] M. A. Krasnosel'skiĭ, Ya. B. Rutickiĭ, Convex functions and Orlicz spaces, P. Noordhoff Ltd., Groningen, 1961, xi+249 pp. | MR | MR | Zbl | Zbl

[15] 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. | MR | Zbl

[16] E. V. Tokarev, “O podprostranstvakh nekotorykh simmetrichnykh prostranstv”, Teoriya funktsii, funktsionalnyi analiz i ikh prilozheniya, 24, Vischa shkola, Kharkov, 1975, 156–161 | MR | Zbl

[17] P. N. Dowling, N. Randrianantoanina, “Asymptotically isometric copies of $\ell^\infty$ in Banach spaces and a theorem of Bessaga and Pełczyński”, Proc. Amer. Math. Soc., 128:11 (2000), 3391–3397 | DOI | MR | Zbl