The largest linear space of operators satisfying the Daugavet equation in 𝐿₁

Shvydkoy, R.

doi:10.1090/S0002-9939-01-06179-2

The largest linear space of operators satisfying the Daugavet equation in $L_{1}$
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Proc. Amer. Math. Soc. 130 (2002), 773-777 Request permission

Abstract:

We find the largest linear space of bounded linear operators on $L_1(\Omega )$ that, being restricted to any $L_1(A)$, $A\subset \Omega$, satisfy the Daugavet equation.

References

V. F. Babenko and S. A. Pičugov, On a property of compact operators in the space of integrable functions, Ukrain. Mat. Zh. 33 (1981), no. 4, 491–492 (Russian). MR 627725
I. K. Daugavet, A property of completely continuous operators in the space $C$, Uspehi Mat. Nauk 18 (1963), no. 5 (113), 157–158 (Russian). MR 0157225
Leonard E. Dor, On projections in $L_{1}$, Ann. of Math. (2) 102 (1975), no. 3, 463–474. MR 420244, DOI 10.2307/1971039
James R. Holub, Daugavet’s equation and operators on $L^1(\mu )$, Proc. Amer. Math. Soc. 100 (1987), no. 2, 295–300. MR 884469, DOI 10.1090/S0002-9939-1987-0884469-8
Vladimir M. Kadets, Roman V. Shvidkoy, Gleb G. Sirotkin, and Dirk Werner, Banach spaces with the Daugavet property, Trans. Amer. Math. Soc. 352 (2000), no. 2, 855–873. MR 1621757, DOI 10.1090/S0002-9947-99-02377-6
G. Y. Lozanovsky, On almost integral operators in KB-spaces, Vestnik Leningrad. Univ., 7 (1966), 35-44.
Anatoliĭ M. Plichko and Mikhail M. Popov, Symmetric function spaces on atomless probability spaces, Dissertationes Math. (Rozprawy Mat.) 306 (1990), 85. MR 1082412
R. V. Shvidkoy, Geometric aspects of the Daugavet property, J. Funct. Anal. 176 (2000), 198–212.