id: 05947273 dt: b an: 05947273 au: Gustafson, Karl ti: Antieigenvalue analysis. With applications to numerical analysis, wavelets, statistics, quantum mechanics, finance and optimization. so: Hackensack, NJ: World Scientific (ISBN 978-981-4366-28-1/hbk; 978-981-4366-29-8/ebook). xiv, 244~p. \$~89.00, \sterling~59.00/hbk; \$~116.00, \sterling~77.00/ebook (2012). py: 2012 pu: Hackensack, NJ: World Scientific la: EN cc: ut: antieigenvalue; antieigenvector; angle of the operator; operator trigonometry ci: li: http://ebooks.worldscinet.com/ISBN/9789814366298/toc.shtml ab: The theory of antieigenvalues and antieigenvectors was created by Karl Gustafson in the period 1966‒1969. If $x$ is an eigenvector of an operator $A$, associated with the eigenvalue $λ,$ then the vector direction of $x$ is preserved under the action of $A,$ subject to a scale change $λ.$ Roughly speaking, antieigenvectors $x$ are those vectors most turned by $A,$ and the corresponding antieigenvalue is the cosine of the maximal turning angle. The maximal turning angle, denoted by $ϕ\left( A\right) ,$ is called the angle of the operator. So the author created also an operator trigonometry, based on the following $\min-\max$ theorem: Let $A$ be a strongly accretive bounded operator on a Hilbert space. Then $$1-\cos ^{2}ϕ\left( A\right) = \sup_{\left\Vert x\right\Vert \leq 1} \inf_{ε\in \mathbb{R}} \left\Vert \left( εA-I\right) x\right\Vert ^{2} = \inf_{ε\in \mathbb{R}}\sup_{\left\Vert x\right\Vert \leq 1} \left\Vert \left( εA-I\right) x\right\Vert ^{2}.$$ An elementary example, discussed in the Introduction, is concerned with a symmetric positive definite matrix $A$ having eigenvalues $0<λ_{1}\leq λ_{2}\leq \cdots \leq λ_{n}.$ Then the antieigenvectors are $$\pm \sqrt{\frac{λ_{n}}{λ_{1}+λ_{n}}}x_{1}+\sqrt{\frac{ λ_{1}}{λ_{1}+λ_{n}}}x_{n},$$ where $x_{1}$ and $x_{n}$ are any norm-one eigenvectors corresponding to $λ_{1},$ respectively $λ_{n}.$ The associated antieigenvalue is $$\cos ϕ\left( A\right) =\frac{2\sqrt{λ_{1}λ_{n}}}{λ_{1}+λ_{n}}.$$ In creating this theory, the author was originally motivated by the study of the following problem: if $A$ is the infinitesimal generator of a contraction semigroup, find conditions on an operator $B$ under which $BA$ is again the infinitesimal generator of a contraction semigroup of operators. This original motivation is discussed in Chapter 2. Chapter 3 presents the essential features of antieigenvalue theory. The next five chapters are devoted to applications to numerical analysis (Chapter 4), wavelets, control, scattering theory (Chapter 5), matrix statistics (Chapter 6), quantum mechanics (Chapter 7), mathematical finance (Chapter 8). Other applications and possible directions of investigation are discussed in Chapter 9. Each chapter contains six exercises for which hints and answers are provided. The first part of the Bibliography contains the contributions of Karl Gustafson and his co-authors. The second part lists important contributions to the theory of other authors. The third part contains supporting books and papers used within the text. rv: Gheorghe Toader (Cluj-Napoca)