Covering a compact set in a Banach space by an operator range of a Banach space with basis

Fonf, V.; Johnson, W.; Plichko, A.; Shevchyk, V.

doi:10.1090/S0002-9947-05-04083-3

Covering a compact set in a Banach space by an operator range of a Banach space with basis
HTML articles powered by AMS MathViewer

by V. P. Fonf, W. B. Johnson, A. M. Plichko and V. V. Shevchyk PDF

Trans. Amer. Math. Soc. 358 (2006), 1421-1434

Abstract:

A Banach space $X$ has the approximation property if and only if every compact set in $X$ is in the range of a one-to-one bounded linear operator from a space that has a Schauder basis. Characterizations are given for $\mathcal {L}_p$ spaces and quotients of $\mathcal {L}_p$ spaces in terms of covering compact sets in $X$ by operator ranges from $\mathcal {L}_p$ spaces. A Banach space $X$ is a $\mathcal {L}_1$ space if and only if every compact set in $X$ is contained in the closed convex symmetric hull of a basic sequence which converges to zero.

References

G. Bennett, L. E. Dor, V. Goodman, W. B. Johnson, and C. M. Newman, On uncomplemented subspaces of $L_{p},$ $1<p<2$, Israel J. Math. 26 (1977), no. 2, 178–187. MR 435822, DOI 10.1007/BF03007667
T. Figiel, Factorization of compact operators and applications to the approximation problem, Studia Math. 45 (1973), 191–210. (errata insert). MR 336294, DOI 10.4064/sm-45-2-191-210
T. Figiel and W. B. Johnson, The approximation property does not imply the bounded approximation property, Proc. Amer. Math. Soc. 41 (1973), 197–200. MR 341032, DOI 10.1090/S0002-9939-1973-0341032-5
V. P. Fonf, One property of families of imbedded Banach spaces, J. Soviet Math. 59 (1992), no. 1, 687–690. Translation of Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. No. 55 (1991), 140–145 [ MR1219949 (94e:46031)]. MR 1157744, DOI 10.1007/BF01102495
V. P. Fonf, On the extension of operational bases in Banach spaces, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 54 (1990), 37–41 (Russian); English transl., J. Soviet Math. 58 (1992), no. 4, 319–322. MR 1080722, DOI 10.1007/BF01097281
V. P. Fonf, W. B. Johnson, G. Pisier, and D. Preiss, Stochastic approximation properties in Banach spaces, Studia Math. 159 (2003), no. 1, 103–119. Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday. MR 2030905, DOI 10.4064/sm159-1-5
Y. Gordon and D. R. Lewis, Absolutely summing operators and local unconditional structures, Acta Math. 133 (1974), 27–48. MR 410341, DOI 10.1007/BF02392140
Wojciech Herer, Stochastic bases in Fréchet spaces, Demonstratio Math. 14 (1981), no. 3, 719–724 (1982). MR 663121
William B. Johnson, Factoring compact operators, Israel J. Math. 9 (1971), 337–345. MR 290133, DOI 10.1007/BF02771684
W. B. Johnson, H. P. Rosenthal, and M. Zippin, On bases, finite dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488–506. MR 280983, DOI 10.1007/BF02771464
J. Lindenstrauss and H. P. Rosenthal, The ${\cal L}_{p}$ spaces, Israel J. Math. 7 (1969), 325–349. MR 270119, DOI 10.1007/BF02788865
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056
N. J. Nielsen and P. Wojtaszczyk, A remark on bases in ${\cal L}_{p}$-spaces with an application to complementably universal ${\cal L}_{\infty }$-spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 249–254 (English, with Russian summary). MR 322484
A. Pełczyński, Any separable Banach space with the bounded approximation property is a complemented subspace of a Banach space with a basis, Studia Math. 40 (1971), 239–243. MR 308753, DOI 10.4064/sm-40-3-239-243
A. Pełczyński, Universal bases, Studia Math. 32 (1969), 247–268. MR 241954, DOI 10.4064/sm-32-3-247-268
A. Pełczyński and H. P. Rosenthal, Localization techniques in $L^{p}$ spaces, Studia Math. 52 (1974/75), 263–289. MR 361729
A. N. Pličko, Choice in Banach space of subspaces with special properties and certain properties of quasicomplements, Funktsional. Anal. i Prilozhen. 15 (1981), no. 1, 82–83 (Russian). MR 609803
Gideon Schechtman, On Pełczyński’s paper “Universal bases” (Studia Math. 32 (1969), 247–268), Israel J. Math. 22 (1975), no. 3-4, 181–184. MR 390730, DOI 10.1007/BF02761587
Stanisław J. Szarek, A Banach space without a basis which has the bounded approximation property, Acta Math. 159 (1987), no. 1-2, 81–98. MR 906526, DOI 10.1007/BF02392555
Paolo Terenzi, A complement to Kreĭn-Mil′man-Rutman theorem, with applications, Istit. Lombardo Accad. Sci. Lett. Rend. A 113 (1979), 341–353 (1981) (English, with Italian summary). MR 622113