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A PSEUDOSPECTRAL MAPPING THEOREM 1. Introduction The MATHEMATICS OF COMPUTATION Volume 72, Number 244, Pages 1841{1854 S 0025-5718(03)01542-4 Article electronically published on May 20, 2003 A PSEUDOSPECTRAL MAPPING THEOREM S.-H. LUI Abstract. The pseudospectrum has become an important quantity for an- alyzing stability of nonnormal systems. In this paper, we prove a mapping theorem for pseudospectra, extending an earlier result of Trefethen. Our re- sult consists of two relations that are sharp and contains the spectral mapping theorem as a special case. Necessary and sufficient conditions for these re- lations to collapse to an equality are demonstrated. The theory is valid for bounded linear operators on Banach spaces. For normal matrices, a special version of the pseudospectral mapping theorem is also shown to be sharp. Some numerical examples illustrate the theory. 1. Introduction The spectrum of a normal matrix paints an accurate picture of the behavior of the matrix. Here, normality refers to the matrix having a complete set of orthogonal eigenvectors. The spectrum of a nonnormal matrix, however, may not be very informative. Thanks largely to the work of Trefethen and his coworkers ([2], [4]), the pseudospectrum has emerged as an appropriate indicator for the stability of nonnormal systems. It has been applied to problems in hydrodynamic instability, turbulence, magnetohydrodynamics, control theory, the iterative solution of linear equations, the numerical solution of differential equations, quantum mechanics, random matrices, etc. See the review articles mentioned above for references. For a matrix A and nonnegative number , recall that the -pseudospectrum of A can be defined as the following closed set in the complex plane: [ Λ(A) ≡ Λ(A + E): kEk≤ Here Λ( ) denotes the spectrum of a matrix and k·kis the matrix 2-norm. When A is a normal matrix, its -pseudospectrum is the union of closed disks of radius with centers at the eigenvalues. For a nonnormal matrix, its -pseudospectrum can be much bigger than this union. The spectral mapping theorem is a fundamental result in functional analysis of great importance. Given a matrix A and a function f which is analytic on an open set containing Λ(A), the theorem asserts that f(Λ(A)) = Λ(f(A)): Received by the editor October 11, 2001 and, in revised form, March 29, 2002. 2000 Mathematics Subject Classification. Primary 15A18, 15A60, 65F15. Key words and phrases. Pseudospectra, eigenvalues, spectral mapping theorem. This work was supported in part by a grant from NSERC. c 2003 American Mathematical Society 1841 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1842 S.-H. LUI In [1], the authors consider sixteen pseudospectral theorems that generalize cor- responding spectral theorems: each -pseudospectral theorem reduces to an eigen- value theorem when = 0. Theorem 14 in this collection states that Λjβj(α + βA)=α+βΛ(A)forα, β complex numbers. When =0,thisisthespectralmap- ping theorem for affine functions. Actually, this result had already appeared earlier in [3]. The goal of this paper is to extend this mapping theorem for pseudospectra to analytic functions. The following is an outline of the paper. In Section 2, the general pseudospectral mapping theorem in the form of two set inclusions is stated and proved. It is shown that the set inclusions reduce to an equality for all matrices iff the mapping is an affine function. A weaker version of the pseudospectral mapping theorem that turns out to have interesting consequences is demonstrated in Section 3. In Section 4, an adaptation of this \weak" theorem for normal matrices is shown to be sharp. This is followed by some numerical experiments which illustrate the theory in the final section. In the remainder of this introduction, we offer two simple examples to illustrate the role of and motivation for studying pseudospectra. First consider a system of ordinary differential equations x0(t)=f(x(t)). Suppose f(0) = 0 and Df(0), the matrix of partial derivatives of f, has eigenvalues whose real parts are all negative. Then it is well understood that 0 is an (asymptotically) stable equilibrium point in the sense of Liapunov. If Df(0) is a highly nonnormal matrix, then the pseudospectrum of Df(0) may protrude far into the right half of the complex plane (assuming that Df(0) has one eigenvalue not too far away from the imaginary axis). While eigenvalue analysis predicts stability, pseudospectral analysis predicts instability which is usually the more acceptable answer unless we are only interested in the ideal system without noise or other errors. Next, consider the system x0(t)=f(x(t),λ), where λ is a real parameter. Sup- pose for all λ in some neighborhood containing 0;f(0,λ)=0andDf(0,λ)has eigenvalues α(λ) iβ(λ), where α and β are real. Suppose further that at λ =0 there is exactly one pair of pure imaginary eigenvalues iβ(0), where β(0) is pos- itive, and dα/dλ =6 0. Then the celebrated Poincare-Andronov-Hopf bifurcation theorem states that there exists a branch of time-periodic solutions in a neighbor- hood of (x; λ)=(0; 0) of period approximately 2π/β(0). However, if Df(0; 0) is highly nonnormal, then a slight perturbation of Df(0; 0)mayleadtonoperiodic solution or to a periodic solution with a period possibly very different from 2π/β(0). 2. Pseudospectral mapping theorem The following is a pseudospectral mapping theorem for complex analytic func- tions. It is sharp in the sense that the pair of functions (φ, ) defining the sizes of the pseudospectra are optimal. Actually, the theorem is an easy consequence of the definitions of these functions. Theorem 2.1. Let A be a matrix and let f be an analytic function defined on D, an open set containing Λ(A).Foreach, s ≥ 0 and sufficiently small, define φ()= sup inffr ≥ 0;f(ζ) 2 Λr(f(A))g ζ2Λ(A) and (s)= sup inffr ≥ 0;z2 f(Λr(A))g: z2Λs(f(A)) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A PSEUDOSPECTRAL MAPPING THEOREM 1843 Then f(Λ(A)) ⊂ Λφ()(f(A)) ⊂ f(Λ (φ())(A)): Proof. We first show that φ is well defined. Let ζ 2 Λ(A). Then ζ 2 Λ(A + E)for some matrix E such that kEk≤. By the spectral mapping theorem, f(ζ) 2 Λ(f(A + E)) = Λ(f(A)+F ); where F = f(A + E) − f(A) and satisfies kF k≤r for some positive r.Wemay take r to be independent of ζ since ζ 2 Λ(A). Thus f(ζ) 2 Λr(f(A)) which implies that the infimum in the definition of φ is taken over a nonempty set and thus φ is well defined and φ() ≤ r. The first set inclusion now follows directly from the definition of φ. Next we show that is well defined. Let z 2 Λs(f(A)). Then z 2 Λ(f(A)+F ) for some matrix F such that kF k≤s.Letz = f(ζ)forsomeζ 2D.(Ifsuchζ does not exist, then z 62 f(D) which is impossible if s is small enough.) Therefore z 2 f(Λr(A)) for some positive r which may be taken to be independent of z 2 Λs(f(A)). Thus the infimum in the definition of is taken over a nonempty set and so is well defined and (s) ≤ r. The second set inclusion now follows directly from the definition of with s = φ(). This completes the proof. Here are some remarks. (1) Note that φ(0) = 0 = (0) and thus the theorem reduces to the usual spectral mapping theorem when =0. (2) If f(z)=α + βz,whereα, β are complex numbers, then φ()= sup inffr ≥ 0,α+ βζ 2 Λ(α + βA + F ); ζ2Λ(A) some F such that kF k≤rg =supinffr ≥ 0,α+ βζ 2 α + βΛ(A + F=β); ζ2Λ(A) some F such that kF k≤rg =supinffr ≥ 0,α+ βζ 2 α + βΛr=jβj(A)g ζ2Λ(A) = jβj: In a similar way, (s)=s=jβj and hence (φ()) = , recovering Theorem 14 in [1]. Here, we have assumed β =0;otherwise,itistriviallytrue.6 (3) By the definitions of φ and , the set inclusions are sharp in the sense that the functions cannot be replaced by smaller functions. (4) It is well known that an equivalent definition of the -pseudospectrum is −1 −1 Λ(A)=fz 2 C; k(z − A) k≥ g; where the norm is taken to be 1 if z 2 Λ(A). It follows that (2.1) φ()= sup k(f(ζ) − f(A))−1k−1 ζ2Λ(A) and (2.2) (s)= sup inf k(ζ − A)−1k−1: 2 −1 z2Λs(f(A)) ζ f (z) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1844 S.-H. LUI The following is a short justification. φ()= supinffr ≥ 0;f(ζ) 2 Λr(f(A))g ζ2Λ(A) =supinffr ≥ 0; k(f(ζ) − f(A))−1k≥r−1g ζ2Λ(A) =supk(f(ζ) − f(A))−1k−1: ζ2Λ(A) Similarly, (s)= sup inffr ≥ 0;z2 f(Λr(A))g z2Λs(f(A)) =supinffr ≥ 0,ζ2 f −1(z); k(ζ − A)−1k≥r−1g z2Λs(f(A)) =sup inf k(ζ − A)−1k−1: 2 −1 z2Λs(f(A)) ζ f (z) For a simple analytic function f and a small matrix A so that f(A)can be explicitly computed, (2.1) and (2.2) are more convenient for numerical computation. For instance, to calculate φ(), since it is a monotone in- creasing function of , we only need to examine points along @Λ(A), the boundary of Λ(A): (2.3) sup k(f(ζ) − f(A))−1k−1: ζ2@Λ(A) (5) The spectral mapping theorem is valid for bounded linear operators on Ba- nach spaces and so is the pseudospectral mapping theorem.
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