Pseudospectrum

In mathematics, the pseudospectrum of an operator is a set containing the spectrum of the operator and the numbers that are "almost" eigenvalues. Knowledge of the pseudospectrum can be particularly useful for understanding non-normal operators and their eigenfunctions.

The ε-pseudospectrum of a matrix A consists of all eigenvalues of matrices which are ε-close to A:

$\Lambda _{\epsilon }(A)=\{\lambda \in \mathbb {C} \mid \exists x\in \mathbb {C} ^{n}\setminus \{0\},\exists E\in \mathbb {C} ^{n\times n}\colon (A+E)x=\lambda x,\|E\|\leq \epsilon \}.$

Numerical algorithms which calculate the eigenvalues of a matrix give only approximate results due to rounding and other errors. These errors can be described with the matrix E.

More generally, for Banach spaces $X,Y$ and operators $A:X\to Y$ , one can define the $\epsilon$-pseudospectrum of $A$ (typically denoted by ${\text{sp}}_{\epsilon }(A)$) in the following way

${\text{sp}}_{\epsilon }(A)=\{\lambda \in \mathbb {C} \mid \|(A-\lambda I)^{-1}\|\geq 1/\epsilon \}.$

where we use the convention that $\|(A-\lambda I)^{-1}\|=\infty$ if $A-\lambda I$ is not invertible.

Bibliography

• Lloyd N. Trefethen and Mark Embree: "Spectra And Pseudospectra: The Behavior of Nonnormal Matrices And Operators", Princeton Univ. Press, ISBN 978-0691119465 (2005).

External links

• Pseudospectra Gateway by Embree and Trefethen