History
Year:
-
Type:
Journal
Book
Article
Please fill in your query. A complete syntax description you will find on the General Help page.
Entropy, compactness and the approximation of operators. (English)
Cambridge Tracts in Mathematics, 98. Cambridge etc.: Cambridge University Press. x, 277 p. \sterling 35.00; \$69.50 (1990). The book of B. Carl and I. Stephani gives an exposition of the theory of entropy numbers, approximation numbers and related quantities which was developed during the last 15 years, in particular. The general procedure is to define operator ideals using these numbers which is the general point of view of A. Pietsch and his working group. The intention of the book is to close the gap between general courses in functional analysis and research papers e.g. in the geometry of Banach spaces where entropy numbers were used in various contexts. It assumes only basic knowledge in functional analysis and approximation theory. The basic concept is the one of entropy numbers of operators$T: X\to Y$$$e\sb k(T)=\inf \{ϵ>0\vert \quad \exists\sb{x\sb 1,...,x\sb{2\sp{n-1}}\in Y} T(B\sb X)\le \cup\sp{2\sp{n- 1}}\sb{i=1}(\{x\sb i\}+ϵB\sb Y)\},$$ a certain inverse of the covering numbes of$T(B\sb X)$by$B\sb Y$-multiples. Chapters 1 and 2 contain the definition and basic results on these numbers, approximation numbers$a\sb n(T)$and related quantities like the entropy moduli$g\sb n(T)$. Chapter 3 surveys the relation between the entropy quantities and the approximation- or Gelfand numbers. They can be viewed as abstract analogues of the inequalities of Bernstein and Jackson in approximation theory. Typical is inequality (3.1) due to Carl, $$\sup\sb{n\in {\bbfN}}e\sb n(T)n\spα\le c\sbα\sup\sb{n\in {\bbfN}}a\sb n(T)n\spα\quad (α>0),$$ which was used by various authors e.g. in connection with Pisier’s generalization of Milman’s ellipsoids (6.5). Another approach yields bounds for$e\sb n(T)$or$g\sb n(T)$in terms of geometric means of the$a\sb k(T)$-sequence. The subject of chapter 4 is a quantitative form of the Riesz theory of compact operators: the eigenvalues of such maps are estimated by entropy- or other s-numbers. A basic inequality of Carl-Triebel states for$T: X\to X$$$(4.2.1) (\prod\sp{n}\sb{i=1}\vert λ\sb i(T)\vert)\sp{1/n}\le g\sb n(T)$$ here$λ\sb i(T)$denotes the sequence of eigenvalues of T. As a generalization of classical spectral radius formula, several asymptotic formulas for (means of) eigenvalues by expressions like$\lim\sb{n}s\sb k(T\sp n)\sp{1/n}$are proved. The main chapter in terms of applications is chapter 5: various compactness results of operators T:$X\to C(K)$, K compact, metric, are derived. One of the main results is (5.6.1), an estimate of Jackson-type $$a\sb{n+1}(T)\le \sup\sb{\Vert x\Vert \le 1}ω(Tx;δ),$$$ω=$modulus of continuity. If$C\spα(K)$denotes the$α$-Hölder continuous functions on K$(0<α<1)$, one has$a\sb n(C\spα(K)\to C(K))\sim ϵ\sb n(K)\spα,$(5.6/7/8). These results, relying on techniques of Carl-Heinrich-Kühn and Timan, yield estimates for the s-numbers of maps factoring as$T: X\to C\spα(K)\to\sp{I}C(K).$Actually, local techniques involving estimates for maps$S: X\to \ell\sp n\sb{\infty}$are shown to be sufficient to prove general results on$T: X\to C(K)\$. Applications to integral operators are given in (5.11) to (5.13). Chapter 6 deals with some applications in the local theory of Banach spaces, to projection constants, distances to Hilbert spaces, p-summing operators, Santaló’s inequality. The book is a welcome addition to the literature on various aspects of the geometry of Banach spaces. The reader should be warned, however, that there are various topics on entropy numbers proved during the last decade which are outside the scope of the book, e.g. the relation to the 1-norm, the duality problem for entropy numbers, the proof of Pisier’s extension of Milman’s ellipsoid result for applications of entropy- and covering numbers in probability in Banach spaces like Dudley’s metric entropy results.
H.König