##
**Tensor norms and operator ideals.**
*(English)*
Zbl 0774.46018

North-Holland Mathematics Studies. 176. Amsterdam: North-Holland. xi, 566 p. (1993).

Tensor products of normed spaces were studied in Grothendieck’s “Résumé” in 1956, the theory of operator ideals was created in the 1960’s and 1970’s, in particular by Pietsch and his school. There is an intimate relation between both concepts, by identifying tensors on \(E^*\otimes F\) with operators from \(E\) to \(F\). Nevertheless, both theories developed quite independently from one another. Both have their own advantages and disadvantages. The book by A. Defant and K. Floret is the first text developing systematically both theories of tensor products and operator ideals and studying the close relationship between them. In applications, of course, both points of view have been used at the same time before, e.g. by G. Pisier in his “Factorization of linear operators and geometry of Banach spaces” [Reg. Conf. Ser. Math. 60, X, 154 p. (1986; Zbl 0588.46010)]. Presenting a unified approach to both topics, the book is a welcome addition to the literature. It can be used as a text at the graduate level.

Chapter I is of introductory character (sections 1-11): the authors discuss bilinear maps, tensor products, operator ideals and study the basic examples, like the \(\varepsilon\) and \(\pi\) tensor products, integral and summing operators, approximation properties and norms related to vector-valued \(L_ p\)-spaces.

The basic theory of tensor products of normed spaces and its relationship to operator ideals is presented in chapter II (sections 12-27). A basic problem for tensor norms in the general question in what sense the norms are determined finite-dimensionally, e.g. by finite-dimensional subspaces or quotients. For this reason, (co) finite hulls of tensor norms are studied, the coincidence of both concepts – total accessibility – is important for duality purposes. The somewhat weaker notion of accessible ideal norms is often sufficient, too, and linked to approximation properties. Important results on the equivalence of tensor norm notions and operator ideal concepts are Theorem 17.5 on the duality of associate tensor norms and maximal operator ideals, Theorem 22.2 on minimal operator ideals and Theorem 17.6 on the cofinite hull of tensor norms. Projective and injective norms or their associates are studied in section 20. Section 14 presents Grothendieck’s inequality in various equivalent formulations; as a consequence, various inequalities relating different tensor norms or ideal norms are given. The book gives many examples of norms, e.g. those of Lapresté-type \(\alpha_{p,q}\) in the tensor case or \((p,q)\)-factorable or \((p,q)\)-dominated operators in the ideal case, or the norms \(\Delta_ p\) induced by Bochner \(L_ p\)-spaces which violate the mapping property; this is the origin of difficulties with vector-valued analoga of scalar results in harmonic analysis but leads to interesting applications e.g. in the case of the Fourier-transform (section 31). In connection with summing or mixing operators, stable measures are studied. In many examples, properties of operators on \(L_ p(E)\)-spaces are considered. Chapter II culminates with the 14 tensor norms of Grothendieck originating from \(\varepsilon\), \(\pi\) by applying the basic operations of adjoint, transpose and injective/projective associates.

Section III contains more specialized topics: factorization of operators through Hilbert spaces, non-accessible tensor norms, mixing operators, tensor stability, Radon-Nikodym property and tensor products and ideals for locally convex spaces.

The book contains a large number of interesting exercises.

Chapter I is of introductory character (sections 1-11): the authors discuss bilinear maps, tensor products, operator ideals and study the basic examples, like the \(\varepsilon\) and \(\pi\) tensor products, integral and summing operators, approximation properties and norms related to vector-valued \(L_ p\)-spaces.

The basic theory of tensor products of normed spaces and its relationship to operator ideals is presented in chapter II (sections 12-27). A basic problem for tensor norms in the general question in what sense the norms are determined finite-dimensionally, e.g. by finite-dimensional subspaces or quotients. For this reason, (co) finite hulls of tensor norms are studied, the coincidence of both concepts – total accessibility – is important for duality purposes. The somewhat weaker notion of accessible ideal norms is often sufficient, too, and linked to approximation properties. Important results on the equivalence of tensor norm notions and operator ideal concepts are Theorem 17.5 on the duality of associate tensor norms and maximal operator ideals, Theorem 22.2 on minimal operator ideals and Theorem 17.6 on the cofinite hull of tensor norms. Projective and injective norms or their associates are studied in section 20. Section 14 presents Grothendieck’s inequality in various equivalent formulations; as a consequence, various inequalities relating different tensor norms or ideal norms are given. The book gives many examples of norms, e.g. those of Lapresté-type \(\alpha_{p,q}\) in the tensor case or \((p,q)\)-factorable or \((p,q)\)-dominated operators in the ideal case, or the norms \(\Delta_ p\) induced by Bochner \(L_ p\)-spaces which violate the mapping property; this is the origin of difficulties with vector-valued analoga of scalar results in harmonic analysis but leads to interesting applications e.g. in the case of the Fourier-transform (section 31). In connection with summing or mixing operators, stable measures are studied. In many examples, properties of operators on \(L_ p(E)\)-spaces are considered. Chapter II culminates with the 14 tensor norms of Grothendieck originating from \(\varepsilon\), \(\pi\) by applying the basic operations of adjoint, transpose and injective/projective associates.

Section III contains more specialized topics: factorization of operators through Hilbert spaces, non-accessible tensor norms, mixing operators, tensor stability, Radon-Nikodym property and tensor products and ideals for locally convex spaces.

The book contains a large number of interesting exercises.

Reviewer: H.König (Kiel)

### MSC:

46B28 | Spaces of operators; tensor products; approximation properties |

46M05 | Tensor products in functional analysis |

47L20 | Operator ideals |

46B07 | Local theory of Banach spaces |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |