| Title:
|
On the complexity of some classes of Banach spaces and non-universality (English) |
| Author:
|
Braga, Bruno M. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
64 |
| Issue:
|
4 |
| Year:
|
2014 |
| Pages:
|
1123-1147 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
These notes are dedicated to the study of the complexity of several classes of separable Banach spaces. We compute the complexity of the Banach-Saks property, the alternating Banach-Saks property, the complete continuous property, and the LUST property. We also show that the weak Banach-Saks property, the Schur property, the Dunford-Pettis property, the analytic Radon-Nikodym property, the set of Banach spaces whose set of unconditionally converging operators is complemented in its bounded operators, the set of Banach spaces whose set of weakly compact operators is complemented in its bounded operators, and the set of Banach spaces whose set of Banach-Saks operators is complemented in its bounded operators, are all non Borel in ${\rm SB}$. At last, we give several applications of those results to non-universality results. (English) |
| Keyword:
|
Banach-Saks operator |
| Keyword:
|
Dunford-Pettis property |
| Keyword:
|
analytic Radon-Nikodym property |
| Keyword:
|
complete continuous property |
| Keyword:
|
Schur property |
| Keyword:
|
unconditionally converging operator |
| Keyword:
|
weakly compact operator |
| Keyword:
|
local structure |
| Keyword:
|
non-universality |
| Keyword:
|
$\ell _p$-Baire sum |
| Keyword:
|
descriptive set theory |
| Keyword:
|
tree |
| MSC:
|
46B20 |
| idZBL:
|
Zbl 06433718 |
| idMR:
|
MR3304802 |
| DOI:
|
10.1007/s10587-014-0157-y |
| . |
| Date available:
|
2015-02-09T17:48:19Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144165 |
| . |
| Reference:
|
[1] Albiac, F., Kalton, N. J.: Topics in Banach Space Theory.Graduate Texts in Mathematics 233 Springer, Berlin (2006). MR 2192298 |
| Reference:
|
[2] Argyros, S. A., Dodos, P.: Genericity and amalgamation of classes of Banach spaces.Adv. Math. 209 (2007), 666-748. Zbl 1109.03047, MR 2296312, 10.1016/j.aim.2006.05.013 |
| Reference:
|
[3] Argyros, S. A., Haydon, R. G.: A hereditarily indecomposable $\mathcal{L}_\infty$-space that solves the scalar-plus-compact problem.Acta Math. 206 (2011), 1-54. Zbl 1223.46007, MR 2784662, 10.1007/s11511-011-0058-y |
| Reference:
|
[4] Bahreini, M., Bator, E., Ghenciu, I.: Complemented subspaces of linear bounded operators.Can. Math. Bull. 55 (2012), 449-461. Zbl 1255.46006, MR 2957262, 10.4153/CMB-2011-097-2 |
| Reference:
|
[5] Beauzamy, B.: Banach-Saks properties and spreading models.Math. Scand. 44 (1979), 357-384. Zbl 0427.46007, MR 0555227, 10.7146/math.scand.a-11818 |
| Reference:
|
[6] Bossard, B.: A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces.Fundam. Math. 172 (2002), 117-152. Zbl 1029.46009, MR 1899225, 10.4064/fm172-2-3 |
| Reference:
|
[7] Bourgain, J., Delbaen, F.: A class of special $\mathcal{L}_\infty$ spaces.Acta Math. 145 (1980), 155-176. MR 0590288, 10.1007/BF02414188 |
| Reference:
|
[8] Diestel, J.: A survey of results related to the Dunford-Pettis property.Proc. Conf. on Integration, Topology, and Geometry in Linear Spaces Contemp. Math. 2 American Mathematical Society (1980), 15-60. Zbl 0571.46013, MR 0621850 |
| Reference:
|
[9] Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators.Cambridge Studies in Advanced Mathematics 43 Cambridge Univ. Press, Cambridge (1995). Zbl 0855.47016, MR 1342297 |
| Reference:
|
[10] J. Diestel, J. J. Uhl, Jr.: Vector Measures.Mathematical Surveys 15 American Mathematical Society, Providence (1977). Zbl 0369.46039, MR 0453964 |
| Reference:
|
[11] Diestel, J., Seifert, C. J.: The Banach-Saks ideal, I. Operators acting on $C(\Omega)$.Commentat. Math. 1 (1978), 109-118. Zbl 0385.46010, MR 0504156 |
| Reference:
|
[12] Dodos, P.: Banach Spaces and Descriptive Set Theory: Selected Topics.Lecture Notes in Mathematics 1993 Springer, Berlin (2010). Zbl 1215.46002, MR 2598479 |
| Reference:
|
[13] Fakhoury, H.: Sur les espaces de Banach ne contenant pas $\ell^1(\mathbb N)$.French Math. Scand. 41 (1977), 277-289. MR 0500085, 10.7146/math.scand.a-11720 |
| Reference:
|
[14] Farnum, N. R.: The Banach-Saks theorem in $C(S)$.Can. J. Math. 26 (1974), 91-97. Zbl 0253.54026, MR 0367636, 10.4153/CJM-1974-009-9 |
| Reference:
|
[15] Girardi, M.: Dentability, trees, and Dunford-Pettis operators on $L_1$.Pac. J. Math. 148 (1991), 59-79 correction ibid. 157 389-394 (1993). MR 1091530, 10.2140/pjm.1991.148.59 |
| Reference:
|
[16] Huang, S.-Z., Neerven, J. M. A. M. van: $B$-convexity, the analytic Radon-Nikodym property, and individual stability of $C_0$-semigroups.J. Math. Anal. Appl. 231 (1999), 1-20. MR 1676753, 10.1006/jmaa.1998.6211 |
| Reference:
|
[17] James, R. C.: Unconditional bases and the Radon-Nikodym property.Stud. Math. 95 (1990), 255-262. Zbl 0744.46010, MR 1060728, 10.4064/sm-95-3-255-262 |
| Reference:
|
[18] James, R. C.: Structures of Banach spaces: Radon-Nikodym and other properties.General Topology and Modern Analysis Proc. Conf., Riverside, 1980 L. F. McAuley, et al. Academic Press, New York (1981), 347-363. MR 0619061 |
| Reference:
|
[19] James, R. C.: Bases and reflexivity of Banach spaces.Ann. Math. (2) 52 (1950), 518-527. Zbl 0039.12202, MR 0039915, 10.2307/1969430 |
| Reference:
|
[20] Johnson, W. B., (eds.), J. Lindenstrauss: Handbook of the Geometry of Banach Spaces. Vol. 1.Elsevier, Amsterdam (2001). MR 1863689 |
| Reference:
|
[21] Kechris, A. S.: Classical Descriptive Set Theory.Graduate Texts in Mathematics 156 Springer, Berlin (1995). Zbl 0819.04002, MR 1321597 |
| Reference:
|
[22] Ostrovskii, M. I.: Topologies on the set of all subspaces of a Banach space and related questions of Banach space geometry.Quaest. Math. 17 (1994), 259-319. MR 1290670, 10.1080/16073606.1994.9631766 |
| Reference:
|
[23] Partington, J. R.: On the Banach-Saks property.Math. Proc. Camb. Philos. Soc. 82 (1977), 369-374. Zbl 0368.46018, MR 0448036, 10.1017/S0305004100054025 |
| Reference:
|
[24] Pełczyński, A.: Universal bases.Stud. Math. 32 (1969), 247-268. Zbl 0185.37401, MR 0241954, 10.4064/sm-32-3-247-268 |
| Reference:
|
[25] Pełczyński, A.: Banach spaces on which every unconditionally converging operator is weakly compact.Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 10 (1962), 641-648. Zbl 0107.32504, MR 0149295 |
| Reference:
|
[26] Puglisi, D.: The position of $\mathcal K(X,Y)$ in $\mathcal L(X,Y)$.Glasg. Math. J. 56 (2014), 409-417. Zbl 1296.46018, MR 3187907 |
| Reference:
|
[27] Rosenthal, H. P.: A characterization of Banach spaces containing $\ell^1$.Proc. Natl. Acad. Sci. USA 71 (1974), 2411-2413. MR 0358307, 10.1073/pnas.71.6.2411 |
| Reference:
|
[28] Rosenthal, H. P.: On injective Banach spaces and the spaces $L^\infty(\mu)$ for finite measures $\mu$.Acta Math. 124 (1970), 205-248. MR 0257721, 10.1007/BF02394572 |
| Reference:
|
[29] Schlumprech, Th.: Notes on Descriptive Set Theory and Applications to Banach Spaces.Class notes for Reading Course in Spring/Summer (2008). |
| Reference:
|
[30] Tanbay, B.: Direct sums and the Schur property.Turk. J. Math. 22 (1998), 349-354. Zbl 0923.46019, MR 1675081 |
| . |
