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Pseudospectra of linear operators
 SIAM Rev
, 1997
"... Abstract. If a matrix or linear operator A is far from normal, its eigenvalues or, more generally, its spectrum may have little to do with its behavior as measured by quantities such as ‖An ‖ or ‖exp(tA)‖. More may be learned by examining the sets in the complex plane known as the pseudospectra of A ..."
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Cited by 154 (10 self)
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Abstract. If a matrix or linear operator A is far from normal, its eigenvalues or, more generally, its spectrum may have little to do with its behavior as measured by quantities such as ‖An ‖ or ‖exp(tA)‖. More may be learned by examining the sets in the complex plane known as the pseudospectra of A, defined by level curves of the norm of the resolvent, ‖(zI − A) −1‖. Five years ago, the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications. Since that time, analogous computations have been carried out for differential and integral operators. This paper, a companion to the earlier one, presents ten examples, each chosen to illustrate one or more mathematical or physical principles.
Nonselfadjoint harmonic oscillator, compact semigroups and pseudospectra
 J. Operator Theory
"... We provide new information concerning the pseudospectra of the complex harmonic oscillator. Our analysis illustrates two different techniques for getting resolvent norm estimates. The first uses the JWKB method and extends for this particular potential some results obtained recently by E.B. Davies. ..."
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Cited by 21 (2 self)
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We provide new information concerning the pseudospectra of the complex harmonic oscillator. Our analysis illustrates two different techniques for getting resolvent norm estimates. The first uses the JWKB method and extends for this particular potential some results obtained recently by E.B. Davies. The second relies on the fact that the bounded holomorphic semigroup generated by the complex harmonic oscillator is of HilbertSchmidt type in a maximal angular region. In order to show this last property, we deduce a nonselfadjoint version of the classical Mehler’s formula.
Norms of Inverses, Spectra, and Pseudospectra of Large Truncated WienerHopf Operators and Toeplitz Matrices
, 1997
"... . This paper is concerned with WienerHopf integral operators on L p and with Toeplitz operators (or matrices) on l p . The symbols of the operators are assumed to be continuous matrix functions. It is well known that the invertibility of the operator itself and of its associated operator imply ..."
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Cited by 17 (3 self)
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. This paper is concerned with WienerHopf integral operators on L p and with Toeplitz operators (or matrices) on l p . The symbols of the operators are assumed to be continuous matrix functions. It is well known that the invertibility of the operator itself and of its associated operator imply the invertibility of all sufficiently large truncations and the uniform boundedness of the norms of their inverses. Quantitative statements, such as results on the limit of the norms of the inverses, can be proved in the case p = 2 by means of C algebra techniques. In this paper we replace C algebra methods by more direct arguments to determine the limit of the norms of the inverses and thus also of the pseudospectra of large truncations in the case of general p. Contents 1. Introduction 2 2. Structure of Inverses 4 3. Norms of Inverses 7 4. Spectra 15 5. Pseudospectra 17 6. Matrix Case 22 7. Block Toeplitz Matrices 23 8. Pseudospectra of Infinite Toeplitz Matrices 27 References 30 ...
On the pseudospectrum of elliptic quadratic differential operators, preprint
, 2007
"... Abstract. We study the pseudospectrum of a class of nonselfadjoint differential operators. Our work consists in a detailed study of the microlocal properties, which rule the spectral stability or instability phenomena appearing under small perturbations for elliptic quadratic differential operators ..."
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Cited by 12 (3 self)
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Abstract. We study the pseudospectrum of a class of nonselfadjoint differential operators. Our work consists in a detailed study of the microlocal properties, which rule the spectral stability or instability phenomena appearing under small perturbations for elliptic quadratic differential operators. The class of elliptic quadratic differential operators stands for the class of operators defined in the Weyl quantization by complexvalued elliptic quadratic symbols. We establish in this paper a simple necessary and sufficient condition on the Weyl symbol of these operators, which ensures the stability of their spectra. When this condition is violated, we prove that it occurs some strong spectral instabilities for the high energies of these operators, in some regions which can be far away from their spectra. We give a precise geometrical description of them, which explains the results obtained for these operators in some numerical simulations giving the computation of “false eigenvalues ” far from their spectra by algorithms for eigenvalues computing. Key words. Spectral instability, pseudospectrum, semiclassical quasimodes, nonselfadjoint operators, nonnormal operators, condition (Ψ), subellipticity.
LDU Factorization Results For BiInfinite And SemiInfinite Scalar And Block Toeplitz Matrices
, 1996
"... In this article various existence results for the LDUfactorization of semiinfinite and biinfinite scalar and block Toeplitz matrices and numerical methods for computing them are reviewed. Moreover, their application to the orthonormalization of splines is indicated. Both banded and nonbanded Toep ..."
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Cited by 8 (0 self)
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In this article various existence results for the LDUfactorization of semiinfinite and biinfinite scalar and block Toeplitz matrices and numerical methods for computing them are reviewed. Moreover, their application to the orthonormalization of splines is indicated. Both banded and nonbanded Toeplitz matrices are considered. Extensive use is made of matrix polynomial theory. Results on the approximation by the LDUfactorizations of finite sections are discussed. The generalization of the results to the LDUfactorization of multiindex Toeplitz matrices is outlined.
Spatial discretization of Cuntz algebras
 Houston Math. J
"... The (abstract) Cuntz algebra ON is generated by nonunitary isometries and has therefore no intrinsic finiteness properties. To approximate the elements of the Cuntz algebra by finitedimensional objects, we thus consider a spatial discretization of ON by the finite sections method. For we represent ..."
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Cited by 6 (5 self)
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The (abstract) Cuntz algebra ON is generated by nonunitary isometries and has therefore no intrinsic finiteness properties. To approximate the elements of the Cuntz algebra by finitedimensional objects, we thus consider a spatial discretization of ON by the finite sections method. For we represent the Cuntz algebra as a (concrete) algebra of operators on l 2 (Z +) and associate with each operator A in this algebra the sequence (PnAPn) of its finite sections. The goal of this paper is to examine the structure of the C ∗algebra S(ON) which is generated by all sequences of this form. Our main results are the fractality of a suitable restriction of the algebra S(ON) and a necessary and sufficient criterion for the stability of sequences in the restricted algebra. These results are employed to study spectral and pseudospectral approximations of elements of ON. 1
Sensitivity Analysis of Partial Differential Equations With Applications to Fluid Flow
, 2005
"... For over 100 years, researchers have attempted to predict transition to turbulence in fluid flows by analyzing the spectrum of the linearized NavierStokes equations. However, for many simple flows, this approach has failed to match experimental results. Recently, new scenarios for transition have b ..."
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Cited by 3 (0 self)
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For over 100 years, researchers have attempted to predict transition to turbulence in fluid flows by analyzing the spectrum of the linearized NavierStokes equations. However, for many simple flows, this approach has failed to match experimental results. Recently, new scenarios for transition have been proposed that are based on the nonnormality of the linearized operator. These new “mostly linear ” theories have increased our understanding of the transition process, but the role of nonlinearity has not been explored. The main goal of this work is to begin to study the role of nonlinearity in transition. We use model problems to illustrate that small unmodeled disturbances can cause transition through movement or bifurcation of equilibria. We also demonstrate that small wall roughness can lead to transition by causing the linearized system to become unstable. Sensitivity methods are used to obtain important information about the disturbed problem and to illustrate that it is possible to have a precursor to predict transition. Finally, we apply linear feedback control to the model problems to illustrate the power of feedback to delay transition and even relaminarize fully developed chaotic flows.
SEMICLASSICAL HYPOELLIPTIC ESTIMATES WITH A BIG LOSS OF DERIVATIVES
"... Abstract. We study the pseudospectral properties of general pseudodifferential operators around a doubly characteristic point and provide necessary and sufficient conditions for semiclassical hypoelliptic a priori estimates with a big loss of derivatives to hold. 1. ..."
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Abstract. We study the pseudospectral properties of general pseudodifferential operators around a doubly characteristic point and provide necessary and sufficient conditions for semiclassical hypoelliptic a priori estimates with a big loss of derivatives to hold. 1.