##
**Banach-Mazur distances and finite-dimensional operator ideals.**
*(English)*
Zbl 0721.46004

Pitman Monographs and Surveys in Pure and Applied Mathematics, 38. Harlow: Longman Scientific & Technical; New York: John Wiley & Sons, Inc. xii, 395 p. (1989).

At the present time, some of the most active research in the theory of Banach spaces concentrates on the area known as the local theory. This theory developed from papers of A. Grothendieck published in the mid- 1950s. It was also influenced in an essential way by A. Dvortzky’s theorem on almost spherical sections of convex bodies and a circle of geometric ideas around it. In the early 1970s, the notions of type and cotype of Banach spaces, and their relations with probability in Banach spaces and operator theory, provided a strong impulse forwards. The present development of the local theory gives it an important place in the theory of Banach spaces and its applications.

The local theory investigates the structure of finite-dimensional Banach spaces and the relations between infinite-dimensional Banach spaces and their finite-dimensional subspaces. The evaluation of various isometric invariants of finite-dimensional Banach spaces, the so-called parameters, plays a central part in this approach. Isomorphic invariants of infinite- dimensional Banach spaces are derived from the asymptotic behaviour of parameters of their finite-dimensional subspaces, as the dimensions grow to infinity.

There is an intimate connection between the local structure of Banach spaces and properties of finite-rank operators acting between them. Consequently, many invariants of the local theory come from the theory of operator ideals. Many important theorems on operators acting between special Banach spaces, combined with general arguments about operator ideals, broaden the scope of Banach space theory in an essential way, and also provide powerful tools for the study of the structure of Banach spaces. On the other hand, methods from the theory of Banach spaces are used successfully to obtain quantitative estimates in operator theory which extend classical results.

The Banach-Mazur distance between Banach spaces plays a fundamental role in the study of local properties of Banach spaces. Methods developed in the study of Banach-Mazur distances and other techniques of the local theory have contributed to the solution of several long-standing problems concerning Banach spaces, and have also been applied to other areas of mathematics such as operator theory, probability and harmonic analysis.

This book concentrates mainly on operator-theoretic aspects of the local theory of Banach spaces. It gives a systematic presentation of the basic theory of operator ideals and their metric properties, and of results and methods concerning Banach-Mazur distances and parameters of finite- dimensional Banach spaces.

The book is addressed to specialists and to students who want to gain a working knowledge of operator ideal theory and methods in the local theory of Banach spaces. The reader is assumed to have a basic knowledge of functional analysis and probability. The book will also be a convenient reference source for researchers in the field.

Chapter 1 is a preliminary one. The rest of the book can be divided into three parts. The first part, chapters 2 to 5, deals with operator ideals. Our general approach comes from the theory of Banach spaces rather than from operator ideal theory. The choice of material is determined by its usefulness for the study of Banach spaces. The basic operator ideals and the fundamental theorems about operators acting between special Banach spaces are discussed in chapter 2. Chapter 3 is devoted to a study of geometric properties of ellipsoids which are defined in finite- dimensional Banach spaces, and which satisfy certain extremal conditions expressed in terms of ideal norms. Chapter 4 enters deeper into the theory of (q,r)-summing operators. It is concerned, in particular, with operators on C(K)-spaces and with some metric properties of (q,2)-summing norms on operators of finite rank. This in turn leads to the concept of dimensional gradation and cogradation of an operator ideal, introduced in chapter 5. This can be quite useful for the theory of Banach spaces.

The second part, chapters 6 to 8, concentrates on variants related to the distances to the euclidean space \(\ell^ n_ 2\) and to \(\ell^ n_{\infty}\). In particular, in section 29, a complete proof of the Lindenstrauss-Tzafriri theorem on complemented subspaces is given, yielding the best-known estimates. In chapter 8 the projection constant, the Gordon-Lewis constant and other related invariants are discussed.

The third part, chapters 9 to 12, is devoted to the study of distances between finite-dimensional Banach spaces, and provides a comprehensive account of the subject. It makes critical use of the existence and geometric properties of extremal ellipsoids defined by ideal norms. Many constructions are based upon the phenomenon of concentration of measure on the Euclidean sphere. A related approach, based on results about Gaussian processes, leads to estimates for norms of random orthogonal factorizations. These estimates are used to investigate Banach-Mazur distances within various classes of spaces.

Each chapter ends with notes which contain comments and references.

The local theory investigates the structure of finite-dimensional Banach spaces and the relations between infinite-dimensional Banach spaces and their finite-dimensional subspaces. The evaluation of various isometric invariants of finite-dimensional Banach spaces, the so-called parameters, plays a central part in this approach. Isomorphic invariants of infinite- dimensional Banach spaces are derived from the asymptotic behaviour of parameters of their finite-dimensional subspaces, as the dimensions grow to infinity.

There is an intimate connection between the local structure of Banach spaces and properties of finite-rank operators acting between them. Consequently, many invariants of the local theory come from the theory of operator ideals. Many important theorems on operators acting between special Banach spaces, combined with general arguments about operator ideals, broaden the scope of Banach space theory in an essential way, and also provide powerful tools for the study of the structure of Banach spaces. On the other hand, methods from the theory of Banach spaces are used successfully to obtain quantitative estimates in operator theory which extend classical results.

The Banach-Mazur distance between Banach spaces plays a fundamental role in the study of local properties of Banach spaces. Methods developed in the study of Banach-Mazur distances and other techniques of the local theory have contributed to the solution of several long-standing problems concerning Banach spaces, and have also been applied to other areas of mathematics such as operator theory, probability and harmonic analysis.

This book concentrates mainly on operator-theoretic aspects of the local theory of Banach spaces. It gives a systematic presentation of the basic theory of operator ideals and their metric properties, and of results and methods concerning Banach-Mazur distances and parameters of finite- dimensional Banach spaces.

The book is addressed to specialists and to students who want to gain a working knowledge of operator ideal theory and methods in the local theory of Banach spaces. The reader is assumed to have a basic knowledge of functional analysis and probability. The book will also be a convenient reference source for researchers in the field.

Chapter 1 is a preliminary one. The rest of the book can be divided into three parts. The first part, chapters 2 to 5, deals with operator ideals. Our general approach comes from the theory of Banach spaces rather than from operator ideal theory. The choice of material is determined by its usefulness for the study of Banach spaces. The basic operator ideals and the fundamental theorems about operators acting between special Banach spaces are discussed in chapter 2. Chapter 3 is devoted to a study of geometric properties of ellipsoids which are defined in finite- dimensional Banach spaces, and which satisfy certain extremal conditions expressed in terms of ideal norms. Chapter 4 enters deeper into the theory of (q,r)-summing operators. It is concerned, in particular, with operators on C(K)-spaces and with some metric properties of (q,2)-summing norms on operators of finite rank. This in turn leads to the concept of dimensional gradation and cogradation of an operator ideal, introduced in chapter 5. This can be quite useful for the theory of Banach spaces.

The second part, chapters 6 to 8, concentrates on variants related to the distances to the euclidean space \(\ell^ n_ 2\) and to \(\ell^ n_{\infty}\). In particular, in section 29, a complete proof of the Lindenstrauss-Tzafriri theorem on complemented subspaces is given, yielding the best-known estimates. In chapter 8 the projection constant, the Gordon-Lewis constant and other related invariants are discussed.

The third part, chapters 9 to 12, is devoted to the study of distances between finite-dimensional Banach spaces, and provides a comprehensive account of the subject. It makes critical use of the existence and geometric properties of extremal ellipsoids defined by ideal norms. Many constructions are based upon the phenomenon of concentration of measure on the Euclidean sphere. A related approach, based on results about Gaussian processes, leads to estimates for norms of random orthogonal factorizations. These estimates are used to investigate Banach-Mazur distances within various classes of spaces.

Each chapter ends with notes which contain comments and references.

### MSC:

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 |

47L20 | Operator ideals |

46B20 | Geometry and structure of normed linear spaces |

47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |

47L10 | Algebras of operators on Banach spaces and other topological linear spaces |