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## Studia Mathematica

1997 | 125 | 3 | 289-303
Tytuł artykułu

### Tauberian operators on $L_1(μ)$ spaces

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EN
Abstrakty
EN
We characterize tauberian operators $T:L_1(μ) → Y$ in terms of the images of disjoint sequences and in terms of the image of the dyadic tree in $L_1[0,1]$. As applications, we show that the class of tauberian operators is stable under small norm perturbations and that its perturbation class coincides with the class of all weakly precompact operators. Moreover, we prove that the second conjugate of a tauberian operator $T:L_1(μ) → Y$ is also tauberian, and the induced operator $T̃: L_1(μ)**/L_1(μ) → Y**/Y$ is an isomorphism into. Also, we show that $L_1(μ)$ embeds isomorphically into the quotient of $L_1(μ)$ by any of its reflexive subspaces.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
289-303
Opis fizyczny
Daty
wydano
1997
otrzymano
1996-12-02
poprawiono
1997-05-12
Twórcy
autor
Bibliografia
• [1] T. Alvarez and M. González, Some examples of tauberian operators, Proc. Amer. Math. Soc. 111 (1991), 1023-1027.
• [2] B. Beauzamy, Introduction to Banach Spaces and their Geometry, North-Holland Math. Stud. 68, North-Holland, 1985.
• [3] W. J. Davis, T. Figiel, W. B. Johnson and A. Pe/lczy/nski, Factoring weakly compact operators, J. Funct. Anal. 17 (1974), 311-327.
• [4] J. Diestel, Sequences and Series in Banach Spaces, Springer, 1984.
• [5] J. Diestel and J. Uhl, Vector Measures, Math. Surveys 15, Amer. Math. Soc., 1977.
• [6] P. Enflo and T. W. Starbird, Subspaces of $L^1$ containing $L^1$, Studia Math. 65 (1979), 203-225.
• [7] M. González, Properties and applications of tauberian operators, Extracta Math. 5 (1990), 91-107.
• [8] M. González and A. Martínez-Abejón, Supertauberian operators and perturbations, Arch. Math. (Basel) 64 (1995), 423-433.
• [9] M. González and V. M. Onieva, Characterizations of tauberian operators and other semigroups of operators, Proc. Amer. Math. Soc. 108 (1990), 399-405.
• [10] R. H. Herman, Generalizations of weakly compact operators, Trans. Amer. Math. Soc. 132 (1968), 377-386.
• [11] R. C. James, Characterizations of reflexivity, Studia Math. 23 (1964), 205-216.
• [12] R. C. James, Weak compactness and reflexivity, Israel J. Math. 2 (1964), 101-119.
• [13] M. I. Kadec and A. Pełczyński, Bases, lacunary sequences and complemented subspaces in $L_p$, Studia Math. 21 (1962), 161-176.
• [14] N. Kalton and A. Wilansky, Tauberian operators on Banach spaces, Proc. Amer. Math. Soc. 57 (1976), 251-255.
• [15] A. Lebow and M. Schechter, Semigroups of operators and measures of noncompactness, J. Funct. Anal. 7 (1971), 1-26.
• [16] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I. Sequence Spaces, Springer, 1977.
• [17] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II. Function Spaces, Springer, 1979.
• [18] A. Martínez-Abejón, Semigrupos de operadores y ultrapotencias, Ph.D. thesis, Universidad de Cantabria, 1994.
• [19] D. P. Milman and V. D. Milman, The geometry of nested families with empty intersection. The structure of the unit sphere of a non-reflexive space, Mat. Sb. 66 (1965), 109-118 (in Russian); English transl.: Amer. Math. Soc. Transl. 85 (1969).
• [20] R. Neidinger and H. P. Rosenthal, Norm-attainment of linear functionals on subspaces and characterizations of tauberian operators, Pacific J. Math. 118 (1985), 215-228.
• [21] H. P. Rosenthal, On subspaces of $L_p$, Ann. of Math. 97 (1973), 344-373.
• [22] H. P. Rosenthal, Double dual types and the Maurey characterization of Banach spaces containing $ℓ_1$, in: Texas Functional Analysis Seminar 1983-1984 (Austin, Tex.), Longhorn Notes, The Univ. of Texas Press, Austin, Tex., 1984, 1-37.
• [23] H. P. Rosenthal, On wide-(s) sequences and their applications to certain classes of operators, preprint.
• [24] W. Schachermayer, For a Banach space isomorphic to its square the Radon-Nikodým and the Krein-Milman property are equivalent, Studia Math. 81 (1985), 329-339.
• [25] D. G. Tacon, Generalized semi-Fredholm transformations, J. Austral. Math. Soc. Ser. A 34 (1983), 60-70.
• [26] D. G. Tacon, Generalized Fredholm transformations, ibid. 37 (1984), 89-97.
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Bibliografia
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