Operator ideals. (English) Zbl 1012.47001

Johnson, W. B. (ed.) et al., Handbook of the geometry of Banach spaces. Volume 1. Amsterdam: Elsevier. 437-496 (2001); addenda ibid. Vol.2, 1821 (2003).
The paper under review is an excellent, beautiful and up to date survey concerning the history and development of the theory of operator ideals whose roots are in the famous Grothendieck “Résumé” [A. Grothendieck, Bol. Soc. Mat. São Paulo 8, 1-79 (1956; Zbl 0074.32303)] and, as the authors say in the end of the paper, explain “why the Résumé was the germ-cell of ideal theory”.
The paper is a useful tool for beginning readers in order to understand better this powerful tool in modern mathematics, it contains on almost each step nice, interesting and illuminating comments, remarks and connections with some classical results in mathematics.
The contents of the paper are: 1. Basic concepts and closed ideals; 2. Banach and quasi Banach ideals; 3. Approximation numbers; 4. Ultraproducts, maximality and trace duality; 5. Summing operators and some of their relatives; 6. \(L_p\)-factorable operators; 7. Grothendieck’s theorem; 8. Concrete operators; 9. Rademacher type and cotype; 10. UMD operators, Haar type, and uniform convexity; 11. Extension of classical theorems to vector-valued functions; 12. Operator ideals and tensor products, the approximation property.
The References include 193 papers and books beginning with the year 1901.
For the entire collection see [Zbl 0970.46001].


47-02 Research exposition (monographs, survey articles) pertaining to operator theory
47L20 Operator ideals


Zbl 0074.32303