Abstract
The main theme of this paper is a study in some detail of Banach ideals of continuous linear operators between Banach spaces factoring compactly through lp (1≤p<∞) or co, called p-compact and ∞-compact operators respectively. Recently operators of these types have been studied in [4] within the framework of locally convex spaces which are dense subspaces of p-compact projective limits of Banach spaces. These ideals show close resemblance to the ideals of p-nuclear operators-for the case p=∞ they coincide. Analogously to results of Grothendieck concerning continuous linear operators, we consider vector sequence spaces isometric isomorphic to certain spaces of compact linear operators. A representation theorem for p-compact operators is deduced and isometric properties of the ideal norm are treated. The paper also includes some remarks on unconditional convergence and related operator ideals and a representation for the complete ɛ-tensor product\(\ell^{P}\tilde{\bigotimes}_{\varepsilon}E\) (1≤p<∞) is given.
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References
GROTHENDIECK, A.: Sur certaines classes de suites dans les espaces de Banach et le théorème de Dvoretzky — Rogers. Bol. Soc. Mat Sâo Paulo8, 81–110 (1956)
JOHNSON, W.B.: Factoring compact operators. Israel J. Math.9, 337–345 (1971)
KÖTHE, G.: Topological vector spaces I. Berlin-Heidelberg-New York: Springer 1969
LANG, K.: Über spezielle klassen von Schwartz-Räumen. Arch. Math.28, 287–296 (1977)
LINDENSTRAUSS, J.; Tzafriri, L.: Classical Banach spaces I. Berlin-Heidelberg-New York:Springer 1977
PELCZYNSKI, A.: Banach spaces on which every unconditionally converging operator is weakly compact. Bull.Acad.Pol.Sci.Sér.Sci.Math.Astr. et Phys.10, 641–648 (1962)
PERRSON A.; PIETSCH, A.: p-nukleare und p-integrale abbildungen in Banach- Räumen. Studia Math.33, 19–62 (1969)
PIETSCH, A.: Nuclear locally convex spaces. Berlin-Heidelberg-New York: Springer 1972
PIETSCH, A.: l -faktorisierbare operatoren in Banachräumen. Acta Sci Math. (Szeged)31, 117–123 (1970)
RANDTKE, D.: Characterization of precompact maps, Schwartz spaces and nuclear spaces. Trans.Amer.Math.Soc.165, 87–101 (1972)
RANDTKE,D.: Representation theorems for compact operators. Proc.Amer.Math.Soc.37, 481–485 (1973)
RETHERFORD,J.R.; GORDON, Y.; Lewis,D.R.: Banach ideals of operators with applications to the finite dimensional structure of Banach spaces. Israel J.Math13, 348–360 (1972)
SCHAEFER,H.H.: Banach lattices and positive operators. Berlin-Heidelberg-New York: Springer 1974
SINGER,I.: Bases in Banach spaces I. Berlin-Heidelberg-New York: Springer 1970
TERZIOGLU,T.: Remarks on compact and infinite-nuclear mappings. Math. Balk.2, 251–255 (1972)
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Fourie, J., Swart, J. Banach ideals of p-compact operators. Manuscripta Math 26, 349–362 (1979). https://doi.org/10.1007/BF01170259
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DOI: https://doi.org/10.1007/BF01170259