Characterisations of partially continuous, strictly cosingular and φ- type operators

Labuschagne, L. E.

Glasgow mathematical journal, Tome 33 (1991) no. 2, pp. 203-212

Résumé

We will denote the dimension of a subspace M of X by dim M and the codimension of M with respect to X by codxM or simply cod M if there is no danger of confusion. The classes of infinite dimensional and closed infinite codimensional subspaces of X will be denoted by and respectively with F(X) and F(X) denoting the classes of finite dimensional and of finite codimensional subspaces of X respectively. For a subspace M of X we denote the injection of M into X by and the quotient map from X onto the quotient space X/M by . Where there is no danger of confusion we will write JM and QM. The injection of X into its completion will be denoted by Jx. Letting X′ denote the continuous dual of X we remark that since X′ is isometric to ()′, these two spaces will be considered identical where convenient. The orthogonal complements of subsets M ⊂ X in X′ and K ⊂ X′ in X will be denoted by M⊥ and ⊥K respectively; M⊥X and X⊥K will be used if there is danger of confusion.

Détail

DOI : 10.1017/S0017089500008235

Bibliographie

[1] 1.Cross, R. W., Linear transformations of Tauberian type in normed spaces, Note Mat., to appear. Google Scholar

[2] 2.Cross, R. W., Properties of some norm related functions of unbounded linear operators, Math. Z. 199 (1988), 285–302. Google Scholar | DOI

[3] 3.Cross, R. W., Some continuity properties of linear transformations in normed spaces, Glasgow Math. J. 30 (1988), 243–247. Google Scholar | DOI

[4] 4.Cross, R. W., Unbounded linear transformations of upper semi-Fredholm type in normed spaces, Portugal. Math. 47 (1990), 61–79. Google Scholar

[5] 5.Cross, R. W. and Labuschagne, L. E., Characterisations of operators of lower semi-Fredholm type in normed spaces, Preprint, Dept. of Mathematics, Univ. of Cape Town. Google Scholar

[6] 6.Cross, R. W. and Labuschagne, L. E., Partially continuous and semi-continuous linear operators in normed spaces, Expo. Math. 7 (1989), 189–191. Google Scholar

[7] 7.Goldberg, S., Unbounded linear operators (McGraw-Hill, 1966). Google Scholar

[8] 8.Jarchow, H., Locally convex spaces (BG Teubner, Stuttgart, 1981). Google Scholar | DOI

[9] 9.Köthe, G., General linear transformations of locally convex spaces, Math. Ann. 159 (1965), 309–328. Google Scholar | DOI

[10] 10.Labuschagne, L. E., Unbounded strictly cosingular operators and their adjoints, Preprint, Dept. of Mathematics, Univ. of Stellenbosch. Google Scholar

[11] 11.Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces I (Springer-Verlag, 1977). Google Scholar | DOI

[12] 12.Mackey, G., Note on a theorèm of Murray, Bull. Amer. Math. Soc. 52 (1946), 322–325. Google Scholar | DOI

[13] 13.Pelczynski, A., On strictly singular and strictly cosingular operators I, II, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 13 (1965), 31–41. Google Scholar

[14] 14.Robertson, A. P. and Robertson, W. J., Topological vector spaces (2nd ed.) (Cambridge University Press, 1973). Google Scholar

[15] 15.Whitley, R. J., Strictly singular operators and their conjugates, Trans. Amer. Math. Soc. 113 (1964), 252–261. Google Scholar | DOI