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)