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1995 | 113 | 3 | 265-282
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Representing non-weakly compact operators

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EN
Abstrakty
EN
For each S ∈ L(E) (with E a Banach space) the operator R(S) ∈ L(E**/E) is defined by R(S)(x** + E) = S**x** + E(x** ∈ E**). We study mapping properties of the correspondence S → R(S), which provides a representation R of the weak Calkin algebra L(E)/W(E) (here W(E) denotes the weakly compact operators on E). Our results display strongly varying behaviour of R. For instance, there are no non-zero compact operators in Im(R) in the case of $L^1$ and C(0,1), but R(L(E)/W(E)) identifies isometrically with the class of lattice regular operators on $ℓ^2$ for $E = ℓ^2(J)$ (here J is James' space). Accordingly, there is an operator $T ∈ L(ℓ^2(J))$ such that R(T) is invertible but T fails to be invertible modulo $W(ℓ^2(J))$.
Słowa kluczowe
Twórcy
  • Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cantabria, E-39071 Santander, Spain
autor
  • Department of Mathematics, University of Helsinki, P.O.Box 4 (Hallituskatu 15), Fin-00014 University of Helsinki, Finland
  • Department of Mathematics, University of Helsinki, P.O.Box 4 (Hallituskatu 15), Fin-00014 University of Helsinki, Finland
Bibliografia
  • [AB] C. D. Aliprantis and O. Burkinshaw, Positive Operators, Academic Press, 1985.
  • [AG] T. Alvarez and M. González, Some examples of tauberian operators, Proc. Amer. Math. Soc. 111 (1991), 1023-1027.
  • [A] K. Astala, On measures of noncompactness and ideal variations in Banach spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 29 (1980), 1-42.
  • [AT] K. Astala and H.-O. Tylli, Seminorms related to weak compactness and to Tauberian operators, Math. Proc. Cambridge Philos. Soc. 107 (1990), 367-375.
  • [B] J. Bourgain, $H^∞$ is a Grothendieck space, Studia Math. 75 (1983), 193-216.
  • [BK] J. Buoni and A. Klein, The generalized Calkin algebra, Pacific J. Math. 80 (1979), 9-12.
  • [DEJP] W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński, Factoring weakly compact operators, J. Funct. Anal. 17 (1974), 311-327.
  • [DS] N. Dunford and J. T. Schwartz, Linear Operators, Vol. 1, Interscience, 1958.
  • [GJ] D. P. Giesy and R. C. James, Uniformly non-$ℓ^(1)$ and B-convex Banach spaces, Studia Math. 48 (1973), 61-69.
  • [G] M. González, Dual results of factorization for operators, Ann. Acad. Sci. Fenn. A I Math. 18 (1993), 3-11.
  • [GM] M. González and A. Martinón, Operational quantities derived from the norm and measures of noncompactness, Proc. Roy. Irish Acad. 91A (1991), 63-70.
  • [HWW] P. Harmand, D. Werner and W. Werner, M-Ideals in Banach Spaces and Banach Algebras, Lecture Notes in Math. 1547, Springer, 1993.
  • [K] S. V. Kisljakov, On the conditions of Dunford-Pettis, Pełczyński and Grothendieck, Soviet Math. Dokl. 16 (1975), 1616-1621.
  • [LS] A. Lebow and M. Schechter, Semigroups of operators and measures of noncompactness, J. Funct. Anal. 7 (1971), 1-26.
  • [L] D. H. Leung, Banach spaces with property (w), Glasgow Math. J. 35 (1993), 207-217.
  • [LT1] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Lecture Notes in Math. 338, Springer, 1973.
  • [LT2] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I. Sequence Spaces, Ergeb. Math. Grenzgeb. 92, Springer, 1977.
  • [LW] R. J. Loy and G. A. Willis, Continuity of derivations on B(E) for certain Banach spaces E, J. London Math. Soc. 40 (1989), 327-346.
  • [M] M. J. Meyer, On a topological property of certain Calkin algebras, Bull. London Math. Soc. 24 (1992), 591-598.
  • [P1] A. Pełczyński, Banach spaces on which every unconditionally convergent operator is weakly compact, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 10 (1962), 641-648.
  • [P2] A. Pełczyński, On strictly singular and strictly cosingular operators I, II, ibid. 13 (1965), 31-41.
  • [Pf] H. Pfitzner, Weak compactness in the dual of a C*-algebra is determined commutatively, Math. Ann. 298 (1994), 349-371.
  • [Pi] A. Pietsch, Operator Ideals, North-Holland, 1980.
  • [R] F. Räbiger, Absolutstetigkeit und Ordnungsabsolutstetigkeit von Operatoren, Sitzungsber. Heidelberger Akad. Wiss., Springer, 1991.
  • [Re] C. J. Read, Discontinuous derivations on the algebra of bounded operators on a Banach space, J. London Math. Soc. 40 (1989), 305-326.
  • [S] H. H. Schaefer, On the o-spectrum of order bounded operators, Math. Z. 154 (1977), 79-84.
  • [T1] H.-O. Tylli, A spectral radius problem connected with weak compactness, Glasgow Math. J. 35 (1993), 85-94.
  • [T2] H.-O. Tylli, The essential norm of an operator is not self-dual, Israel J. Math., to appear.
  • [V] M. Valdivia, Banach spaces X with X** separable, ibid. 59 (1987), 107-111.
  • [W] L. Weis, Über schwach folgenpräkompakte Operatoren, Arch. Math. (Basel) 30 (1978), 411-417.
  • [WW] L. Weis and M. Wolff, On the essential spectrum of operators on $L^1$, Seminarberichte Tübingen (Sommersemester 1984), 103-112.
  • [Wo] M. Wojtowicz, On the James space J(X) for a Banach space X, Comment. Math. Prace Mat. 23 (1983), 183-188.
  • [Y1] K.-W. Yang, The generalized Fredholm operators, Trans. Amer. Math. Soc. 216 (1976), 313-326.
  • [Y2] K.-W. Yang, Operators invertible modulo the weakly compact operators, Pacific J. Math. 71 (1977), 559-564.
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Bibliografia
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