##
**Spectra and pseudospectra. The behavior of nonnormal matrices and operators.**
*(English)*
Zbl 1085.15009

Princeton, NJ: Princeton University Press (ISBN 0-691-11946-5/hbk). xviii, 606 p. (2005).

The researches of the authors and their colleagues into the effects of small perturbations on nonnormal linear operators have been collected into this lively monograph involving a wide variety of mathematical models from the fields of matrix algebra, ordinary and partial differential equations, and Markov processes.

The authors provide a unifying approach to the subject of small perturbations by introducing the concept of the \(\varepsilon\)-pseudo-spectrum. Given a linear operator \(A\) on a Banach space the resolvent operator is defined to be \(R_\lambda(A)= (\lambda_I-A)^{-1}\) where \(\lambda\) is any complex number. The spectrum of \(A\), \(\sigma(A)\), is the set of \(\lambda\) such that \(R_\lambda (A)\) does not exist or is unbounded. The \(\varepsilon\)-pseudospectrum \(\sigma_\varepsilon(A)\) of \(A\) is the set of \(\lambda\) such that \(\|R_\lambda(A) \|\geq 1/\varepsilon\), \(\varepsilon>0\). If we choose \(\|R_\lambda(A)\|=\infty\) when \(\lambda\in\sigma(A)\) then \(\sigma_\varepsilon(A)\supseteq\sigma(A)\). If \(A+e\) is a small perturbation of \(A\), \(\|E\|<\varepsilon\) then \(\sigma(A+B)\) tend to fill up the \(\varepsilon\)-pseudospectrum \(\sigma_\varepsilon(A)\) as \(E\) is allowed to vary. The eigenvalues in \(\sigma(A+E)\) are called \(\varepsilon\)-pseudoeigenvalues and their eigenvectors \(\varepsilon\)-pseudoeigenvectors. For differential operators where we have eigenfunctions, these are called \(\varepsilon\)-pseudomodes.

After an introductory chapter the book is organized into chapters which may be read independently. Topics include the following: The Toeplitz matrices. Dynamic behavior generated by \(\exp(tA)\) when \(A\) is either an ordinary or partial differential operator. Non existence of solutions of boundary value problems. Onset of stiffness in stiff equations. Spectral properties of the convection-diffusion equation, Airy equation, and Orr-Sommerfeld equation. A model for advenit of turbulence is presented. Growth of error in iterative procedures in large scale numerical computations is treated including important frequently used routines such as Gauss-Seidel and Krylov subspace approximations.

Additional chapters analyse perturbation by random matrices, stability of numerical approximations to partial differential equations, Markov processes, lazer stability, and shuffling of playing cards.

The book has been written at a level to be accessible to a wide audience of students of the applied sciences. The subject matter has been carefully referenced. Many illustrations are provided showing an amazing diversity of spectra end pseudospectra. A detailed chapter is provided for those who wish to generate software to approximate the spectrum and pseudospectrum in a particular application.

The authors provide a unifying approach to the subject of small perturbations by introducing the concept of the \(\varepsilon\)-pseudo-spectrum. Given a linear operator \(A\) on a Banach space the resolvent operator is defined to be \(R_\lambda(A)= (\lambda_I-A)^{-1}\) where \(\lambda\) is any complex number. The spectrum of \(A\), \(\sigma(A)\), is the set of \(\lambda\) such that \(R_\lambda (A)\) does not exist or is unbounded. The \(\varepsilon\)-pseudospectrum \(\sigma_\varepsilon(A)\) of \(A\) is the set of \(\lambda\) such that \(\|R_\lambda(A) \|\geq 1/\varepsilon\), \(\varepsilon>0\). If we choose \(\|R_\lambda(A)\|=\infty\) when \(\lambda\in\sigma(A)\) then \(\sigma_\varepsilon(A)\supseteq\sigma(A)\). If \(A+e\) is a small perturbation of \(A\), \(\|E\|<\varepsilon\) then \(\sigma(A+B)\) tend to fill up the \(\varepsilon\)-pseudospectrum \(\sigma_\varepsilon(A)\) as \(E\) is allowed to vary. The eigenvalues in \(\sigma(A+E)\) are called \(\varepsilon\)-pseudoeigenvalues and their eigenvectors \(\varepsilon\)-pseudoeigenvectors. For differential operators where we have eigenfunctions, these are called \(\varepsilon\)-pseudomodes.

After an introductory chapter the book is organized into chapters which may be read independently. Topics include the following: The Toeplitz matrices. Dynamic behavior generated by \(\exp(tA)\) when \(A\) is either an ordinary or partial differential operator. Non existence of solutions of boundary value problems. Onset of stiffness in stiff equations. Spectral properties of the convection-diffusion equation, Airy equation, and Orr-Sommerfeld equation. A model for advenit of turbulence is presented. Growth of error in iterative procedures in large scale numerical computations is treated including important frequently used routines such as Gauss-Seidel and Krylov subspace approximations.

Additional chapters analyse perturbation by random matrices, stability of numerical approximations to partial differential equations, Markov processes, lazer stability, and shuffling of playing cards.

The book has been written at a level to be accessible to a wide audience of students of the applied sciences. The subject matter has been carefully referenced. Many illustrations are provided showing an amazing diversity of spectra end pseudospectra. A detailed chapter is provided for those who wish to generate software to approximate the spectrum and pseudospectrum in a particular application.

Reviewer: J. B. Butler jun. (Portland)

### MSC:

15A18 | Eigenvalues, singular values, and eigenvectors |

15-02 | Research exposition (monographs, survey articles) pertaining to linear algebra |

47A10 | Spectrum, resolvent |

47A50 | Equations and inequalities involving linear operators, with vector unknowns |

15-04 | Software, source code, etc. for problems pertaining to linear algebra |

34L05 | General spectral theory of ordinary differential operators |

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

65F10 | Iterative numerical methods for linear systems |