Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Pointwise induced operators on $ L\sb{p}$-spaces


Author: Anzelm Iwanik
Journal: Proc. Amer. Math. Soc. 58 (1976), 173-178
MSC: Primary 47B37; Secondary 46E30
DOI: https://doi.org/10.1090/S0002-9939-1976-0412883-6
MathSciNet review: 0412883
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this note we present a characterization of pointwise induced operators on $ {L_p}$-spaces with finite measures, $ 1 \leq p < \infty $. An operator $ P$ is pointwise induced if and only if $ \vert Pu\vert = P\vert u\vert$ and $ P1 = 1$. As an application we obtain a characterization of the linear positive isometries mapping 1 into 1.


References [Enhancements On Off] (What's this?)

  • [1] S. Banach, Théorie des opérations linéaires, Monografie Mat., PWN, Warsaw, 1932; reprint, Chelsea, New York, 1955. MR 17, 175.
  • [2] N. Dunford and J. T. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302. MR 0117523 (22:8302)
  • [3] A. J. Ellis, Extreme positive operators, Quart. J. Math. Oxford Ser. (2) 15 (1964), 342-344. MR 30 #4157. MR 0173950 (30:4157)
  • [4] M. S. Espelie, Multiplicative and extreme positive operators, Pacific J. Math. 48 (1973), 57-66. MR 49 #1212. MR 0336438 (49:1212)
  • [5] J. Lamperti, On the isometries of certain function-spaces, Pacific J. Math. 8 (1958), 459-466. MR 21 #3764. MR 0105017 (21:3764)
  • [6] G. Lumer, Isometries of Orlicz spaces, Bull. Amer. Math. Soc. 68 (1962), 28-30. MR 24 #A1612. MR 0131764 (24:A1612)
  • [7] -, On the isometries of reflexive Orlicz spaces, Ann. Inst. Fourier (Grenoble) 13 (1963), fasc. 1, 99-109. MR 28 #1485; errata, 30, p. 1205. MR 0158259 (28:1485)
  • [8] R. R. Phelps, Extreme positive operators and homomorphisms, Trans. Amer. Math. Soc. 108 (1963), 265-274. MR 27 #6153. MR 0156224 (27:6153)
  • [9] R. Sikorski, On the inducing of homomorphisms by mappings, Fund. Math. 36 (1949), 7-22. MR 11, 166. MR 0031535 (11:166a)
  • [10] -, Boolean algebras, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Band 25, Academic Press, New York; Springer-Verlag, Berlin, 1964. MR 31 #2178. MR 0177920 (31:2178)
  • [11] K. Yoshida, Functional analysis, Springer-Verlag, Berlin and New York, 1968. MR 39 #741. MR 0239384 (39:741)
  • [S] H. H. Schaefer, Banach lattices and positive operators, Springer-Verlag, Berlin and New York, 1974. MR 0423039 (54:11023)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47B37, 46E30

Retrieve articles in all journals with MSC: 47B37, 46E30


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0412883-6
Keywords: $ {L_p}$-space, nonsingular transformation, nonnegative operator, Borel space, quotient Boolean $ \sigma $-algebra
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society