Isometries on 𝐿_{𝑝,1}

Carothers, N. L.; Turett, B.

doi:10.1090/S0002-9947-1986-0849469-6

Isometries on $L_ {p,1}$
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by N. L. Carothers and B. Turett

Trans. Amer. Math. Soc. 297 (1986), 95-103

DOI: https://doi.org/10.1090/S0002-9947-1986-0849469-6

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Abstract:

The extreme points of the sphere of the Lorentz function space ${L_{p,1}}[0,1]$ are computed. As an application, the linear isometries from ${L_{p,1}}$ into itself are completely described.

References

Z. Altshuler, The modulus of convexity of Lorentz and Orlicz sequence spaces, Notes in Banach spaces, Univ. Texas Press, Austin, Tex., 1980, pp. 359–378. MR 606225
B. Bru and H. Heinich, Isométries positives et propriétés ergodiques de quelques espaces de Banach, Ann. Inst. H. Poincaré Sect. B (N.S.) 17 (1981), no. 4, 377–405 (French, with English summary). MR 644353
J. R. Calder and J. B. Hill, A collection of sequence spaces, Trans. Amer. Math. Soc. 152 (1970), 107–118. MR 265913, DOI 10.1090/S0002-9947-1970-0265913-8
N. L. Carothers, Rearrangement invariant subspaces of Lorentz function spaces, Israel J. Math. 40 (1981), no. 3-4, 217–228 (1982). MR 654578, DOI 10.1007/BF02761363

Rearrangement invariant subspaces of Lorentz function spaces

N. L. Carothers and P. H. Flinn, Embedding $l_{p}^{n^{\alpha }}$ in $l^{n}_{p,q}$, Proc. Amer. Math. Soc. 88 (1983), no. 3, 523–526. MR 699425, DOI 10.1090/S0002-9939-1983-0699425-9
J. Creekmore, Type and cotype in Lorentz $L_{pq}$ spaces, Nederl. Akad. Wetensch. Indag. Math. 43 (1981), no. 2, 145–152. MR 707247, DOI 10.1016/1385-7258(81)90143-8
Mahlon M. Day, Normed linear spaces, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21, Springer-Verlag, New York-Heidelberg, 1973. MR 0344849
J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964, DOI 10.1090/surv/015
Israel Halperin, Function spaces, Canad. J. Math. 5 (1953), 273–288. MR 56195, DOI 10.4153/cjm-1953-031-3
G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395
Edwin Hewitt and Karl Stromberg, Real and abstract analysis, Graduate Texts in Mathematics, No. 25, Springer-Verlag, New York-Heidelberg, 1975. A modern treatment of the theory of functions of a real variable; Third printing. MR 0367121
Richard A. Hunt, On $L(p,\,q)$ spaces, Enseign. Math. (2) 12 (1966), 249–276. MR 223874
W. B. Johnson, B. Maurey, G. Schechtman, and L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 19 (1979), no. 217, v+298. MR 527010, DOI 10.1090/memo/0217
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367, DOI 10.1007/978-3-662-35347-9
G. G. Lorentz, Some new functional spaces, Ann. of Math. (2) 51 (1950), 37–55. MR 33449, DOI 10.2307/1969496
Richard O’Neil, Integral transforms and tensor products on Orlicz spaces and $L(p,\,q)$ spaces, J. Analyse Math. 21 (1968), 1–276. MR 626853, DOI 10.1007/bf02787670
H. L. Royden, Real analysis, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1963. MR 0151555