Pointwise induced operators on 𝐿_{𝑝}-spaces

Iwanik, Anzelm

doi:10.1090/S0002-9939-1976-0412883-6

Pointwise induced operators on $L_{p}$-spaces
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Proc. Amer. Math. Soc. 58 (1976), 173-178 Request permission

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 $|Pu| = P|u|$ and $P1 = 1$. As an application we obtain a characterization of the linear positive isometries mapping 1 into 1.

References

Théorie des opérations linéaires

17

Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
A. J. Ellis, Extreme positive operators, Quart. J. Math. Oxford Ser. (2) 15 (1964), 342–344. MR 173950, DOI 10.1093/qmath/15.1.342
M. Solveig Espelie, Multiplicative and extreme positive operators, Pacific J. Math. 48 (1973), 57–66. MR 336438
John Lamperti, On the isometries of certain function-spaces, Pacific J. Math. 8 (1958), 459–466. MR 105017
G. Lumer, Isometries of Orlicz spaces, Bull. Amer. Math. Soc. 68 (1962), 28–30. MR 131764, DOI 10.1090/S0002-9904-1962-10686-7
Gunter Lumer, On the isometries of reflexive Orlicz spaces, Ann. Inst. Fourier (Grenoble) 13 (1963), 99–109. MR 158259
R. R. Phelps, Extreme positive operators and homomorphisms, Trans. Amer. Math. Soc. 108 (1963), 265–274. MR 156224, DOI 10.1090/S0002-9947-1963-0156224-6
Roman Sikorski, On the inducing of homomorphisms by mappings, Fund. Math. 36 (1949), 7–22. MR 31535, DOI 10.4064/fm-36-1-7-22
Roman Sikorski, Boolean algebras, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 25, Academic Press, Inc., New York; Springer-Verlag, Berlin-New York, 1964. MR 0177920
Kôsaku Yosida, Functional analysis, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 123, Springer-Verlag New York, Inc., New York, 1968. MR 0239384
Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039