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.
@article{bwmeta1.element.bwnjournal-article-smv113i3p283bwm,
author = {T. Alvarez and R. Cross and A. Gouveia},
title = {Adjoint characterisations of unbounded weakly compact, weakly completely continuous and unconditionally converging operators},
journal = {Studia Mathematica},
volume = {113},
year = {1995},
pages = {283-298},
zbl = {0823.47020},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv113i3p283bwm}
}
Alvarez, T.; Cross, R.; Gouveia, A. Adjoint characterisations of unbounded weakly compact, weakly completely continuous and unconditionally converging operators. Studia Mathematica, Tome 113 (1995) pp. 283-298. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv113i3p283bwm/
[00000] [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.
[00001] [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. | Zbl 0715.47002
[00002] [C1] R. W. Cross, Properties of some norm related functions of unbounded linear operators, Math. Z. 199 (1988), 285-302. | Zbl 0639.47009
[00003] [C2] R. W. Cross, Unbounded linear operators of upper semi-Fredholm type in normed spaces, Portugal. Math. 47 (1990), 61-79.
[00004] [C3] R. W. Cross, -operators are Tauberian, Quaestiones Math. 16 (1993), 129-132.
[00005] [C4] R. W. Cross, Note on some characterisations of unbounded weakly compact operators, ibid., to appear.
[00006] [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). | Zbl 0780.47002
[00007] [C6] R. W. Cross, On a theorem of Kalton and Wilansky concerning Tauberian operators, J. Math. Anal. Appl. 171 (1992), 156-170. | Zbl 0780.47001
[00008] [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.
[00009] [CL1] R. W. Cross and L. E. Labuschagne, Partially continuous and semi-continuous linear operators in normed spaces, Exposition. Math. 7 (1989), 189-191. | Zbl 0692.47021
[00010] [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. | Zbl 0787.47011
[00011] [D] J. Dixmier, Sur un théorème de Banach, Duke Math. J. 15 (1948), 1057-1071.
[00012] [DS] N. Dunford and J. T. Schwartz, Linear Operators, Part I, Interscience, New York, 1958.
[00013] [F] V. Fonf, Private communication.
[00014] [G] S. Goldberg, Unbounded Linear Operators, McGraw-Hill, New York, 1966. | Zbl 0148.12501
[00015] [Gv] A. I. Gouveia, Unbounded linear operators in seminormed spaces, Univ. Cape Town Thesis Reprints 9/1990.
[00016] [HM] J. Howard and K. Melendez, Characterizing operators by their first and second adjoint, Bull. Inst. Math. Acad. Sinica 5 (1977), 129-134. | Zbl 0355.47003
[00017] [K] G. Köthe, General linear transformation of locally convex spaces, Math. Ann. 159 (1965), 309-328. | Zbl 0134.11501
[00018] [L] L. E. Labuschagne, Characterisations of partially continuous, strictly cosingular and type operators, Glasgow Math. J. 33 (1991), 203-212. | Zbl 0744.47012
[00019] [LT] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Vol. I, Ergeb. Math. Grenzgeb. 92, Springer, 1977.