Geodesic
CHARACTERIZATIONS OF STRICTLY SINGULAR AND STRICTLY COSINGULAR OPERATORS BY PERTURBATION CLASSES
Glasgow mathematical journal, Tome 54 (2012) no. 1, pp. 87-96

Voir la notice de l'article provenant de la source Cambridge University Press

We consider a class of operators that contains the strictly singular operators and it is contained in the perturbation class of the upper semi-Fredholm operators PΦ+. We show that this class is strictly contained in PΦ+, solving a question of Friedman. We obtain similar results for the strictly cosingular operators and the perturbation class of the lower semi-Fredholm operators PΦ−. We also characterize in terms of PΦ+ and in terms of PΦ−. As a consequence, we show that and are the biggest operator ideals contained in PΦ+ and PΦ−, respectively.
DOI : 10.1017/S0017089511000346
Mots-clés : 47A53 
AIENA, PIETRO; GONZÁLEZ, MANUEL; MARTÍNEZ-ABEJÓN, ANTONIO. CHARACTERIZATIONS OF STRICTLY SINGULAR AND STRICTLY COSINGULAR OPERATORS BY PERTURBATION CLASSES. Glasgow mathematical journal, Tome 54 (2012) no. 1, pp. 87-96. doi: 10.1017/S0017089511000346
@article{10_1017_S0017089511000346,
     author = {AIENA, PIETRO and GONZ\'ALEZ, MANUEL and MART\'INEZ-ABEJ\'ON, ANTONIO},
     title = {CHARACTERIZATIONS {OF} {STRICTLY} {SINGULAR} {AND} {STRICTLY} {COSINGULAR} {OPERATORS} {BY} {PERTURBATION} {CLASSES}},
     journal = {Glasgow mathematical journal},
     pages = {87--96},
     year = {2012},
     volume = {54},
     number = {1},
     doi = {10.1017/S0017089511000346},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089511000346/}
}
TY  - JOUR
AU  - AIENA, PIETRO
AU  - GONZÁLEZ, MANUEL
AU  - MARTÍNEZ-ABEJÓN, ANTONIO
TI  - CHARACTERIZATIONS OF STRICTLY SINGULAR AND STRICTLY COSINGULAR OPERATORS BY PERTURBATION CLASSES
JO  - Glasgow mathematical journal
PY  - 2012
SP  - 87
EP  - 96
VL  - 54
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089511000346/
DO  - 10.1017/S0017089511000346
ID  - 10_1017_S0017089511000346
ER  - 
%0 Journal Article
%A AIENA, PIETRO
%A GONZÁLEZ, MANUEL
%A MARTÍNEZ-ABEJÓN, ANTONIO
%T CHARACTERIZATIONS OF STRICTLY SINGULAR AND STRICTLY COSINGULAR OPERATORS BY PERTURBATION CLASSES
%J Glasgow mathematical journal
%D 2012
%P 87-96
%V 54
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089511000346/
%R 10.1017/S0017089511000346
%F 10_1017_S0017089511000346

[1] 1.Aiena, P., Fredholm and local spectral theory, with application to multipliers (Kluwer, Dordrecht, 2004). Google Scholar

[2] 2.Aiena, P. and González, M., On inessential and improjective operators, Studia Math. 131 (1998), 271–287. Google Scholar

[3] 3.Aiena, P., González, M. and Martinón, A., On the perturbation classes of semi-Fredholm operators, Glasgow Math. J. 45 (2003), 91–95. Google Scholar | DOI

[4] 4.Albiac, F. and Kalton, N., Topics in Banach space theory, Graduate Texts in Math., Vol. 233 (Springer, New York, 2006). Google Scholar

[5] 5.Avilés, A., Cabello, F., Castillo, J. M. F., González, M. and Moreno, Y., On separably injective Banach spaces. Preprint. Google Scholar

[6] 6.Castillo, J. M. F. and Plichko, A., Banach spaces in various positions, J. Funct. Anal. 259 (2010), 2098–2138. Google Scholar | DOI

[7] 7.Friedman, T. L., Relating strictly singular operators to the condition X < Y mod() and resulting perturbations, Analysis (Munich) 22 (2002), 347–354. Google Scholar

[8] 8.Giménez, J., González, M. and Martí nez-Abejón, A., Perturbation of semi-Fredholm operators on products of Banach spaces, J. Operator Theory (to appear). Google Scholar

[9] 9.González, M., The perturbation classes problem in Fredholm theory, J. Funct. Anal. 200 (2003), 65–70. Google Scholar | DOI

[10] 10.González, M. and Martí nez-Abejón, A., Tauberian operators, Operator Theory: Advances and Applications, Vol. 194 (Birkhäuser, Basel, 2010). Google Scholar | DOI

[11] 11.González, M., Martí nez-Abejón, A. and Salas-Brown, M., Perturbation classes for semi-Fredholm operators on subprojective and superprojective spaces, Ann. Acad. Sci. Fennicae Math. (to appear). Google Scholar

[12] 12.González, M. and Salas-Brown, M., Perturbation classes for semi-Fredholm operators in L (μ)-spaces, J. Math. Anal. Appl. 370 (2010), 11–17. Google Scholar | DOI

[13] 13.Gowers, W. T. and Maurey, B., The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), 851–874. Google Scholar | DOI

[14] 14.Kato, T., Perturbation theory for nullity, deficiency and other quantities of linear operators, J. d'Analyse Math. 6 (1958), 261–322. Google Scholar | DOI

[15] 15.Pełczyński, A., On strictly singular and strictly cosingular operators I. Strictly singular and strictly cosingular operators in C(S)-spaces, Bull. Acad. Polon. Sci. 13 (1965), 31–36. Google Scholar

[16] 16.Pełczyński, A., On strictly singular and strictly cosingular operators II. Strictly singular and strictly cosingular operators in L(ν)-spaces, Bull. Acad. Polon. Sci. 13 (1965), 37–41. Google Scholar

[17] 17.Pietsch, A., Operator ideals (North-Holland, Amsterdam, 1980). Google Scholar

[18] 18.Taylor, A. E. and Lay, D. C., Introduction to functional analysis, 2nd ed. (Wiley, New York, 1980). Google Scholar

[19] 19.Vladimirskii, J. I., Strictly cosingular operators, Soviet Math. Doklady 8 (1967), 739–740. Google Scholar

Cité par Sources :