Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-23T21:38:14.275Z Has data issue: false hasContentIssue false
  • English
  • Français

Improjective operators and ideals in a category of Banach spaces

Published online by Cambridge University Press:  09 April 2009

E. Tarafdar
E. Tarafdar
Affiliation:
School of Mathematical Sciences, Flinders University, South Australia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Kato [3] has introduced a class of operators called strictly singular operators. These operators have many properties in common with compact operators. In fact the concept of a strictly singular operator is an extension of the concept of a compact operator. Kato has proved that if X and X′ are Banach Spaces, then the singular operators of X into X′ forms a closed subspace of the space of bounded linear operators of X into X′ and if X = X′, then these operators forms a two-sided ideal in the ring of bounded linear operators on X. He has also shown that the Riers-Schauder theorem holds for the spectrum of a strictly singular operator. Gohberg, Feldman and Markus [23] have treated the same class of operators with an equivalent definition.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 14 , Issue 3 , November 1972 , pp. 274 - 292
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Pelczynski, A., ‘Projections in certain Banacyh space’, Studia Math. 19 (1960), 209–228.CrossRefGoogle Scholar
[2]Kato, T., Perturbation theory for linear operators (Springer-Verlag, Berlin, 1966).Google Scholar
[3]Kato, T., ‘Perturbation theory for nullity, deficiency and other quantities of linear opeators,’ J. Analyse Math. 6 (1958), 261–322.CrossRefGoogle Scholar
[4]Whitley, R. J., ‘Strictly singular operators and their conjugates’, Trans. Amer. Math. Soc. 113 (1964), 252–261.CrossRefGoogle Scholar
[5]Goldberg, S. and Thorpe, E., ‘On some open questions concerning strictly singular operators’, Proc. Amer. Math. Soc. 14 (1963), 334–336.CrossRefGoogle Scholar
[6]Pelczynski, A., ‘On strictly singular and strictly co-singular operators I and II’, Bull. Acad. Polon. Sci. 13 No. 1 (1965), 31–41.Google Scholar
[7]Grothendiek, A., ‘Sur les applications linéaires faiblement compactes d'espaces du type C(K)’, Canad. J. Math. 5 (1953), 129–173.CrossRefGoogle Scholar
[8]Banach, S. and Mazur, S., ‘Zur theorie der linearen dimension’, Studia Math. 4 (1933), 100–112.CrossRefGoogle Scholar
[9]Day, M., Normed linear spaces (Springer, Berlin, 1958).Google Scholar
[10]Philips, R. S.. ‘On linear transformations’, Trans. Amer. Math. Soc. 48 (1940), 516–541.CrossRefGoogle Scholar
[11]Taylor, A., Introduction to functional analysis (Wiley, New York, 1958).Google Scholar
[12]James, R. C., ‘Separable conjugate spaces,’ Pacific J. Math. 10 (1960), 563–571.CrossRefGoogle Scholar
[13]Dunford, N. and Schwartz, J., Linear operators, vol. I (Interscience, New York, 1958).Google Scholar
[14]Pelczynski, A. and Semadeni, Z., ‘Spaces of continuous functions (III)’, Studia Math. 18 (1959), 211–222.CrossRefGoogle Scholar
[15]Sobczyk, A., ‘Projection of the space (m) on its subspace (c 0)’, Bull. Amer. Math. Soc. 47 (1941), 938–947.CrossRefGoogle Scholar
[16]Banach, S., Théorie des opérations linéaires (Warszawa, 1932).Google Scholar
[17]Zaanen, A. C., Linear analysis (North Holland Publicashing Co., Amsterdam, 1953).Google Scholar
[18]Lacey, E. and Whitley, R. J., ‘Conditions under which all the bounded linear operators are compact’, Math. Annalen 158 (1965), 1–5.CrossRefGoogle Scholar
[19]Lacey, H. E. and Morris, P. D., ‘Continuous linear operators on spaces of continuous functions’, Proc. Amer. Math. Soc. 17 No. 4 (1966) 848–853.CrossRefGoogle Scholar
[20]Freyd, P.. Abelian categories, an introduction to the theory of functors (Harper and Row, New York, 1964).Google Scholar
[21]Isbell, J., ‘Subobjects, adequacy, completeness and categories of algebras’, Rozprawy Mat. 38 (1963), 1–32.Google Scholar
[22]Kelly, G. M., ‘On the radical of a category’, J. Austral. Math. Soc. 4 (1964), 299–307.CrossRefGoogle Scholar
[23]Sulgeifer, E. G., ‘On the general theory of radicals in categories’, Mat. Sb. 51 (93) (1960), 487–500; also in Am. Math. Transl., Ser. 2, vol. 59, 150–162.Google Scholar
[24]Sulgeifer, E. G., ‘The lattice of ideals of an object of a category I, II’, Mat. Sb. 54 (96) (1961), 335–344; also in Am. Math. Transl., Ser. 2, vol. 59, 163–190.Google Scholar
[25]Schatten, R., Norm ideals of completely continuous operators, (Springer-Verlag, Berlin, 1960).CrossRefGoogle Scholar
[26]Calkin, J. W., ‘Two-sided ideals and congreunces in the ring of bounded linear operators in Hilbert space’, Annals of Mathematics 42, No. 4 (1941), 839–873.CrossRefGoogle Scholar
[27]Gohberg, I. C., Markus, A. S. and Feldman, I. A., ‘Normally solvable operators and ideals associated with them’, Izv. Moldavsk. Fil. Akad. Nauk SSSR 10, No. 76 (1960), 51–69 (Russian).Google Scholar