Adjoint characterisations of unbounded weakly compact, weakly completely continuous and unconditionally converging operators

T. Alvarez; R. Cross; A. Gouveia

Studia Mathematica (1995)

  • Volume: 113, Issue: 3, page 283-298
  • ISSN: 0039-3223

Abstract

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Characterisations are obtained for the following classes of unbounded linear operators between normed spaces: weakly compact, weakly completely continuous, and unconditionally converging operators. Examples of closed unbounded operators belonging to these classes are exhibited. A sufficient condition is obtained for the weak compactness of T' to imply that of T.

How to cite

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Alvarez, T., Cross, R., and Gouveia, A.. "Adjoint characterisations of unbounded weakly compact, weakly completely continuous and unconditionally converging operators." Studia Mathematica 113.3 (1995): 283-298. <http://eudml.org/doc/216175>.

@article{Alvarez1995,
abstract = {Characterisations are obtained for the following classes of unbounded linear operators between normed spaces: weakly compact, weakly completely continuous, and unconditionally converging operators. Examples of closed unbounded operators belonging to these classes are exhibited. A sufficient condition is obtained for the weak compactness of T' to imply that of T.},
author = {Alvarez, T., Cross, R., Gouveia, A.},
journal = {Studia Mathematica},
keywords = {unbounded linear operators between normed spaces; weakly compact; weakly completely continuous; unconditionally converging operators; weak compactness},
language = {eng},
number = {3},
pages = {283-298},
title = {Adjoint characterisations of unbounded weakly compact, weakly completely continuous and unconditionally converging operators},
url = {http://eudml.org/doc/216175},
volume = {113},
year = {1995},
}

TY - JOUR
AU - Alvarez, T.
AU - Cross, R.
AU - Gouveia, A.
TI - Adjoint characterisations of unbounded weakly compact, weakly completely continuous and unconditionally converging operators
JO - Studia Mathematica
PY - 1995
VL - 113
IS - 3
SP - 283
EP - 298
AB - Characterisations are obtained for the following classes of unbounded linear operators between normed spaces: weakly compact, weakly completely continuous, and unconditionally converging operators. Examples of closed unbounded operators belonging to these classes are exhibited. A sufficient condition is obtained for the weak compactness of T' to imply that of T.
LA - eng
KW - unbounded linear operators between normed spaces; weakly compact; weakly completely continuous; unconditionally converging operators; weak compactness
UR - http://eudml.org/doc/216175
ER -

References

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  1. [AO] T. Alvarez and V. Onieva, A note on three-space ideals of Banach spaces, in: Proccedings of the Tenth Spanish-Portuguese Conference on Mathematics, III (Murcia, 1985), Univ. Murcia, Murcia, 1985, 251-254. 
  2. [BKS] V. I. Bogachev, B. Kirchheim and W. Schachermeyer, Continuous restrictions of linear maps between Banach spaces, Acta Univ. Carolin.-Math. Phys. 30 (2) (1989), 31-35. Zbl0715.47002
  3. [C1] R. W. Cross, Properties of some norm related functions of unbounded linear operators, Math. Z. 199 (1988), 285-302. Zbl0639.47009
  4. [C2] R. W. Cross, Unbounded linear operators of upper semi-Fredholm type in normed spaces, Portugal. Math. 47 (1990), 61-79. 
  5. [C3] R. W. Cross, F + -operators are Tauberian, Quaestiones Math. 16 (1993), 129-132. 
  6. [C4] R. W. Cross, Note on some characterisations of unbounded weakly compact operators, ibid., to appear. 
  7. [C5] R. W. Cross, Linear transformations of Tauberian type in normed spaces, Note Mat. 10 (1990), Suppl. No. 1, 193-203 (volume dedicated to the memory of Professor Gottfried M. Köthe). Zbl0780.47002
  8. [C6] R. W. Cross, On a theorem of Kalton and Wilansky concerning Tauberian operators, J. Math. Anal. Appl. 171 (1992), 156-170. Zbl0780.47001
  9. [C7] R. W. Cross, Adjoints of non-densely defined linear operators, in: Aportaciones Mat., volume dedicated to the memory of Prof. Victor Onieva, Univ. Cantabria, Santander, 1991, 117-136. 
  10. [CL1] R. W. Cross and L. E. Labuschagne, Partially continuous and semi-continuous linear operators in normed spaces, Exposition. Math. 7 (1989), 189-191. Zbl0692.47021
  11. [CL2] R. W. Cross and L. E. Labuschagne, Characterisations of operators of lower semi-Fredholm type in normed spaces, Quaestiones Math. 15 (1992), 151-173. Zbl0787.47011
  12. [D] J. Dixmier, Sur un théorème de Banach, Duke Math. J. 15 (1948), 1057-1071. 
  13. [DS] N. Dunford and J. T. Schwartz, Linear Operators, Part I, Interscience, New York, 1958. 
  14. [F] V. Fonf, Private communication. 
  15. [G] S. Goldberg, Unbounded Linear Operators, McGraw-Hill, New York, 1966. Zbl0148.12501
  16. [Gv] A. I. Gouveia, Unbounded linear operators in seminormed spaces, Univ. Cape Town Thesis Reprints 9/1990. 
  17. [HM] J. Howard and K. Melendez, Characterizing operators by their first and second adjoint, Bull. Inst. Math. Acad. Sinica 5 (1977), 129-134. Zbl0355.47003
  18. [K] G. Köthe, General linear transformation of locally convex spaces, Math. Ann. 159 (1965), 309-328. Zbl0134.11501
  19. [L] L. E. Labuschagne, Characterisations of partially continuous, strictly cosingular and ϕ - type operators, Glasgow Math. J. 33 (1991), 203-212. Zbl0744.47012
  20. [LT] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Vol. I, Ergeb. Math. Grenzgeb. 92, Springer, 1977. 

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