M ( r , s ) -ideals of compact operators

Rainis Haller; Marje Johanson; Eve Oja

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 3, page 673-693
  • ISSN: 0011-4642

Abstract

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We study the position of compact operators in the space of all continuous linear operators and its subspaces in terms of ideals. One of our main results states that for Banach spaces X and Y the subspace of all compact operators 𝒦 ( X , Y ) is an M ( r 1 r 2 , s 1 s 2 ) -ideal in the space of all continuous linear operators ℒ ( X , Y ) whenever 𝒦 ( X , X ) and 𝒦 ( Y , Y ) are M ( r 1 , s 1 ) - and M ( r 2 , s 2 ) -ideals in ℒ ( X , X ) and ℒ ( Y , Y ) , respectively, with r 1 + s 1 / 2 > 1 and r 2 + s 2 / 2 > 1 . We also prove that the M ( r , s ) -ideal 𝒦 ( X , Y ) in ℒ ( X , Y ) is separably determined. Among others, our results complete and improve some well-known results on M -ideals.

How to cite

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Haller, Rainis, Johanson, Marje, and Oja, Eve. "$M(r,s)$-ideals of compact operators." Czechoslovak Mathematical Journal 62.3 (2012): 673-693. <http://eudml.org/doc/246277>.

@article{Haller2012,
abstract = {We study the position of compact operators in the space of all continuous linear operators and its subspaces in terms of ideals. One of our main results states that for Banach spaces $X$ and $Y$ the subspace of all compact operators $\mathcal \{K\}(X,Y)$ is an $M(r_1 r_2, s_1 s_2)$-ideal in the space of all continuous linear operators $\mathcal \{L\}(X,Y)$ whenever $\mathcal \{K\}(X,X)$ and $\mathcal \{K\}(Y,Y)$ are $M(r_1,s_1)$- and $M(r_2,s_2)$-ideals in $\mathcal \{L\}(X,X)$ and $\mathcal \{L\}(Y,Y)$, respectively, with $r_1+s_1/2>1$ and $r_2+s_2/2>1$. We also prove that the $M(r,s)$-ideal $\mathcal \{K\}(X,Y)$ in $\mathcal \{L\}(X,Y)$ is separably determined. Among others, our results complete and improve some well-known results on $M$-ideals.},
author = {Haller, Rainis, Johanson, Marje, Oja, Eve},
journal = {Czechoslovak Mathematical Journal},
keywords = {$M(r,s)$-ideal and $M$-ideal of compact operators; property $M^\ast (r,s)$; compact approximation property; -ideal; -ideal; property ; compact approximation property},
language = {eng},
number = {3},
pages = {673-693},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$M(r,s)$-ideals of compact operators},
url = {http://eudml.org/doc/246277},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Haller, Rainis
AU - Johanson, Marje
AU - Oja, Eve
TI - $M(r,s)$-ideals of compact operators
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 3
SP - 673
EP - 693
AB - We study the position of compact operators in the space of all continuous linear operators and its subspaces in terms of ideals. One of our main results states that for Banach spaces $X$ and $Y$ the subspace of all compact operators $\mathcal {K}(X,Y)$ is an $M(r_1 r_2, s_1 s_2)$-ideal in the space of all continuous linear operators $\mathcal {L}(X,Y)$ whenever $\mathcal {K}(X,X)$ and $\mathcal {K}(Y,Y)$ are $M(r_1,s_1)$- and $M(r_2,s_2)$-ideals in $\mathcal {L}(X,X)$ and $\mathcal {L}(Y,Y)$, respectively, with $r_1+s_1/2>1$ and $r_2+s_2/2>1$. We also prove that the $M(r,s)$-ideal $\mathcal {K}(X,Y)$ in $\mathcal {L}(X,Y)$ is separably determined. Among others, our results complete and improve some well-known results on $M$-ideals.
LA - eng
KW - $M(r,s)$-ideal and $M$-ideal of compact operators; property $M^\ast (r,s)$; compact approximation property; -ideal; -ideal; property ; compact approximation property
UR - http://eudml.org/doc/246277
ER -

References

top
  1. Ausekle, J., Oja, E., 10.1007/BF02384767, Ark. Mat. 36 (1998), 233-239. (1998) Zbl1037.47504MR1650589DOI10.1007/BF02384767
  2. Cabello, J. C., Nieto, E., 10.1216/rmjm/1181071823, Rocky Mt. J. Math. 28 (1998), 61-93. (1998) Zbl0936.46014MR1639829DOI10.1216/rmjm/1181071823
  3. Cabello, J. C., Nieto, E., 10.4064/sm-129-2-185-196, Stud. Math. 129 (1998), 185-196. (1998) Zbl0913.46019MR1608091DOI10.4064/sm-129-2-185-196
  4. Cabello, J. C., Nieto, E., On M -type structures and the fixed point property, Houston J. Math. 26 (2000), 549-560. (2000) Zbl0984.46009MR1811941
  5. Cabello, J. C., Nieto, E., Oja, E., 10.1006/jmaa.1997.5888, J. Math. Anal. Appl. 220 (1998), 334-348. (1998) Zbl0917.47040MR1613976DOI10.1006/jmaa.1997.5888
  6. Cho, C.-M., Johnson, W. B., A characterization of subspaces X of l p for which K ( X ) is an M -ideal in L ( X ) , Proc. Am. Math. Soc. 93 (1985), 466-470. (1985) MR0774004
  7. Feder, M., Saphar, P., 10.1007/BF02757132, Isr. J. Math. 21 (1975), 38-49. (1975) Zbl0325.47028MR0377591DOI10.1007/BF02757132
  8. Godefroy, G., Saphar, P. D., 10.1215/ijm/1255988868, Ill. J. Math. 32 (1988), 672-695. (1988) Zbl0631.46015MR0955384DOI10.1215/ijm/1255988868
  9. Haller, R., Johanson, M., Oja, E., M ( r , s ) -inequality for 𝒦 ( X , Y ) in ℒ ( X , Y ) , Acta Comment. Univ. Tartu. Math. 11 (2007), 69-76. (2007) MR2391972
  10. Haller, R., Oja, E., 10.1007/s000130050114, Arch. Math. 69 (1997), 227-233. (1997) Zbl0914.46019MR1464553DOI10.1007/s000130050114
  11. Harmand, P., Werner, D., Werner, W., M -ideals in Banach Spaces and Banach Algebras. Lecture Notes in Mathematics, Vol. 1547, Springer Berlin (1993). (1993) MR1238713
  12. Heinrich, S., The reflexivity of the Banach space L ( E , F ) , Funkcional. Anal. i Prilož. 8 (1974), 97-98 Russian. (1974) Zbl0295.46040MR0342991
  13. Hennefeld, J., 10.1512/iumj.1979.28.28065, Indiana Univ. Math. J. 28 (1979), 927-934. (1979) Zbl0464.46020MR0551156DOI10.1512/iumj.1979.28.28065
  14. John, K., Werner, D., 10.1023/A:1022433018736, Czech. Math. J. 50 (125) (2000), 51-57. (2000) Zbl1040.46020MR1745458DOI10.1023/A:1022433018736
  15. Johnson, J., 10.1016/0022-1236(79)90042-9, J. Funct. Anal. 32 (1979), 304-311. (1979) Zbl0412.47024MR0538857DOI10.1016/0022-1236(79)90042-9
  16. Kahre, Ü., Kirikal, L., Oja, E., 10.1006/jmaa.2000.7413, J. Math. Anal. Appl. 259 (2001), 439-452. (2001) Zbl0997.46016MR1842070DOI10.1006/jmaa.2000.7413
  17. Kalton, N. J., 10.1007/BF01432152, Math. Ann. 208 (1974), 267-278. (1974) Zbl0266.47038MR0341154DOI10.1007/BF01432152
  18. Kalton, N. J., Banach spaces for which the ideal of compact operators is an M -ideal, C. R. Acad. Sci. Paris, Sér. I Math. 313 (1991), 509-513. (1991) Zbl0755.46006MR1131865
  19. Kalton, N. J., 10.1215/ijm/1255987254, Ill. J. Math. 37 (1993), 147-169. (1993) Zbl0824.46029MR1193134DOI10.1215/ijm/1255987254
  20. Kalton, N. J., Werner, D., Property ( M ) , M -ideals, and almost isometric structure of Banach spaces, J. Reine Angew. Math. 461 (1995), 137-178. (1995) Zbl0823.46018MR1324212
  21. Kivisoo, K., Oja, E., 10.1090/S0002-9939-05-08267-5, Proc. Am. Math. Soc. 133 (2005), 3485-3496. (2005) Zbl1078.39025MR2163583DOI10.1090/S0002-9939-05-08267-5
  22. Lima, Å., 10.4064/sm-113-3-249-263, Stud. Math. 113 (1995), 249-263. (1995) Zbl0826.46013MR1330210DOI10.4064/sm-113-3-249-263
  23. Lima, Å., Nygaard, O., Oja, E., 10.1007/BF02810673, Isr. J. Math. 119 (2000), 325-348. (2000) Zbl0983.46024MR1802659DOI10.1007/BF02810673
  24. Lima, Å., Oja, E., 10.1017/S144678870001017X, J. Aust. Math. Soc. 77 (2004), 91-110. (2004) Zbl1082.46016MR2069027DOI10.1017/S144678870001017X
  25. Lima, Å., Oja, E., Rao, T. S. S. R. K., Werner, D., 10.1307/mmj/1029005074, Mich. Math. J. 41 (1994), 473-490. (1994) Zbl0823.46023MR1297703DOI10.1307/mmj/1029005074
  26. Nygaard, O., Põldvere, M., 10.4064/sm195-3-4, Stud. Math. 195 (2009), 243-255. (2009) Zbl1192.46011MR2559175DOI10.4064/sm195-3-4
  27. Oja, E., Strong uniqueness of the extension of linear continuous functionals according to the Hahn-Banach theorem, Mat. Zametki 43 (1988), 237-246, 302 Russian translation in Math. Notes 43 (1988), 134-139. (1988) MR0939524
  28. Oja, E., Dual de l'espace des opérateurs linéaires continus, C. R. Acad. Sci. Paris, Sér. I Math. 309 (1989), 983-986 French. (1989) Zbl0684.47025MR1054748
  29. Oja, E., Extensions of Functionals and the Structure of the Space of Continuous Linear Operators, Tartu Univ. Publ. Tartu (1991), Russian. (1991) MR1114543
  30. Oja, E., On M -ideals of compact operators and Lorentz sequence spaces, Eesti Tead. Akad. Toim., Füüs. Mat. 40 (1991), 31-36. (1991) Zbl0804.46028MR1124516
  31. Oja, E., A note on M -ideals of compact operators, Tartu Ülikooli Toimetised 960 (1993), 75-92. (1993) Zbl1214.46005MR1231939
  32. Oja, E., 10.1112/S002557930001202X, Mathematika 44 (1997), 120-132. (1997) Zbl0878.46013MR1464382DOI10.1112/S002557930001202X
  33. Oja, E., 10.1090/S0002-9939-98-04600-0, Proc. Am. Math. Soc. 126 (1998), 2747-2753. (1998) Zbl0899.46014MR1469429DOI10.1090/S0002-9939-98-04600-0
  34. Oja, E., 10.1016/S0764-4442(99)80433-9, C. R. Acad. Sci. Paris, Sér. I Math. 328 (1999), 1167-1170 French. (1999) Zbl0934.46012MR1701379DOI10.1016/S0764-4442(99)80433-9
  35. Oja, E., 10.1090/S0002-9947-00-02521-6, Trans. Am. Math. Soc. 352 (2000), 2801-2823. (2000) Zbl0954.46010MR1675226DOI10.1090/S0002-9947-00-02521-6
  36. Oja, E., Põldvere, M., On subspaces of Banach spaces where every functional has a unique norm-preserving extension, Stud. Math. 117 (1996), 289-306. (1996) Zbl0854.46014MR1373851
  37. Oja, E., Põldvere, M., 10.1017/S0308210500019375, Proc. R. Soc. Edinb., Sect. A 129 (1999), 1251-1262. (1999) Zbl0938.46018MR1728530DOI10.1017/S0308210500019375
  38. Oja, E., Zolk, I., 10.1017/S0308210507001266, Proc. R. Soc. Edinb., Sect. A 139 (2009), 551-565. (2009) Zbl1180.46013MR2506787DOI10.1017/S0308210507001266
  39. Põldvere, M., 10.1216/rmjm/1181069389, Rocky Mt. J. Math. 36 (2006), 1651-1663. (2006) MR2285307DOI10.1216/rmjm/1181069389
  40. Werner, D., 10.1006/jath.1994.1002, J. Approximation Theory 76 (1994), 21-30. (1994) Zbl0797.41019MR1257062DOI10.1006/jath.1994.1002

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