Eigenvalues and s-numbers. (Licensed ed.).

*(English)*Zbl 0615.47019
Cambridge Studies in Advanced mathematics, 13. Cambridge etc.: Cambridge University Press. 360 p. (Orig. Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig) (1987).

The well-written treatise by Pietsch gives a comprehensive survey on recent and old results about the distribution of the eigenvalues of abstract and concrete compact operators in Banach spaces and related topics like traces and determinants. In a way, it is a supplement to his monomorph ”Operator ideals” (1978; Zbl 0399.47039).

Following preliminaries on sequence and function spaces, operator ideals and interpolation theory, chapters 1 and 2 contain the necessary results on the class of absolutely summing operators and the s-number ideals. These classes form the main abstract operators for which the decay of eigenvalues is studied (in chapter 3). The s-numbers are introduced axiomatically, the central interest are the (examples of the) approximation numbers and Weyl-numbers. Chapter 3 contains the quantitative version of the Riesz spectral theory for these operators: after treating general Riesz operators and the principle of related operators, Pietsch derives the eigenvalue theorem for Weyl-number ideals (the main abstract result) in 3.6. As corollaries he finds results for the decay of the eigenvalues of absolutely summing, nuclear operators and other concrete operator ideals. Also, generalities on the ”eigenvalue type” of an operator ideal are discussed.

Applications to matrix operators and integral operators are found in chapters 5 and 6. In particular, the optimal summability results are given for Hille-Tamarkin type infinite matrices and kernels (in 5.3. and 6.2.) as well as weakly singular integral operators. Section 6.4. concerns integral operators whose kernels belongs to Besov-space-valued Besov-space, i.e. is a smooth function of two variables. The optimal eigenvalue type of these maps is found using a sequence space characterization of Besov-spaces due to Ciesielski-Figiel and sequence space estimates for ”Besov-matrices” (of 5.4.). Applications to the optimal decay of the Fourier coefficients of certain classes of smooth functions follow in section 6.5.

In Chapter 4, Pietsch gives a new and abstract treatment of traces and determinants of operators in Banach spaces. Traces \(\tau\) and determinants \(\delta\) are introduced axiomatically on an operator ideal \({\mathfrak A}\), requiring e.g. for a trace the property \(\tau (XT)=\tau (TX)\) for all \(T\in {\mathfrak A}\) and continuous X, and similarly for a determinant \(\delta (Id+XT)=\delta (Id+TX)\). A Fredholm determinant theory is developed using these notions. It is shown that there is a one- to-one correspondence between these abstract traces and determinants. Not every operator ideal supports a continuous trace; even the nuclear operators do not. However, it is shown for certain s-number ideals that they support a continuous trace which is even a spectral trace, i.e. the trace in these cases equals the sum of the eigenvalues. In chapter 5 and 6, traces are studied for matrix and integral operators. In general an operator ideal may support more than one continuous trace, as shown in the appendix by an ultrafilter construction (due to Kalton).

The last chapter 7 is a very nice historical survey on the origin of the basic notions of the book (eigenvalues, traces, determinants, matrices, integral operators) as well as of the basic results. Clearly, Pietsch puts a lot of work into this valuable addition of the book. This is also reflected by the comprehensive list of references which include the old and classical as well as the new papers and books.

The book may be considered as a reference work as well as a graduate course on a branch of modern functional analysis.

Following preliminaries on sequence and function spaces, operator ideals and interpolation theory, chapters 1 and 2 contain the necessary results on the class of absolutely summing operators and the s-number ideals. These classes form the main abstract operators for which the decay of eigenvalues is studied (in chapter 3). The s-numbers are introduced axiomatically, the central interest are the (examples of the) approximation numbers and Weyl-numbers. Chapter 3 contains the quantitative version of the Riesz spectral theory for these operators: after treating general Riesz operators and the principle of related operators, Pietsch derives the eigenvalue theorem for Weyl-number ideals (the main abstract result) in 3.6. As corollaries he finds results for the decay of the eigenvalues of absolutely summing, nuclear operators and other concrete operator ideals. Also, generalities on the ”eigenvalue type” of an operator ideal are discussed.

Applications to matrix operators and integral operators are found in chapters 5 and 6. In particular, the optimal summability results are given for Hille-Tamarkin type infinite matrices and kernels (in 5.3. and 6.2.) as well as weakly singular integral operators. Section 6.4. concerns integral operators whose kernels belongs to Besov-space-valued Besov-space, i.e. is a smooth function of two variables. The optimal eigenvalue type of these maps is found using a sequence space characterization of Besov-spaces due to Ciesielski-Figiel and sequence space estimates for ”Besov-matrices” (of 5.4.). Applications to the optimal decay of the Fourier coefficients of certain classes of smooth functions follow in section 6.5.

In Chapter 4, Pietsch gives a new and abstract treatment of traces and determinants of operators in Banach spaces. Traces \(\tau\) and determinants \(\delta\) are introduced axiomatically on an operator ideal \({\mathfrak A}\), requiring e.g. for a trace the property \(\tau (XT)=\tau (TX)\) for all \(T\in {\mathfrak A}\) and continuous X, and similarly for a determinant \(\delta (Id+XT)=\delta (Id+TX)\). A Fredholm determinant theory is developed using these notions. It is shown that there is a one- to-one correspondence between these abstract traces and determinants. Not every operator ideal supports a continuous trace; even the nuclear operators do not. However, it is shown for certain s-number ideals that they support a continuous trace which is even a spectral trace, i.e. the trace in these cases equals the sum of the eigenvalues. In chapter 5 and 6, traces are studied for matrix and integral operators. In general an operator ideal may support more than one continuous trace, as shown in the appendix by an ultrafilter construction (due to Kalton).

The last chapter 7 is a very nice historical survey on the origin of the basic notions of the book (eigenvalues, traces, determinants, matrices, integral operators) as well as of the basic results. Clearly, Pietsch puts a lot of work into this valuable addition of the book. This is also reflected by the comprehensive list of references which include the old and classical as well as the new papers and books.

The book may be considered as a reference work as well as a graduate course on a branch of modern functional analysis.

Reviewer: H.König

##### MSC:

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

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

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

47L05 | Linear spaces of operators |

45C05 | Eigenvalue problems for integral equations |