## Spectral properties of random non-self-adjoint matrices and operators (2001)

Citations: | 8 - 4 self |

### BibTeX

@MISC{Davies01spectralproperties,

author = {E. B. Davies},

title = {Spectral properties of random non-self-adjoint matrices and operators},

year = {2001}

}

### OpenURL

### Abstract

We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from self-adjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also describe a stochastic family of bounded operators in infinite dimensions for almost all of which the eigenvectors generate a dense linear subspace, but the eigenvalues do not determine the spectrum. Our results imply that the spectrum of the non-self-adjoint Anderson model changes suddenly as one passes to the infinite volume limit.

### Citations

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(Show Context)
Citation Context ... 0. The norm of a particular spectral projection is also called the condition number of the eigenvalue, and is known to measure how unstable the eigenvalue is under small perturbations of the matrix, =-=[1, 23, 24]-=-,[16, sect. 11.2]. We emphasize that if the norm of the spectral projection is very large the instability of the eigenvalue it intrinsic: it does not depend on the particular method of computing it. T... |

137 |
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Citation Context ...of Aω are not uniformly bounded. This implies that the eigenvectors cannot form a basis. An example of Zabzyk which has some similarities to those of the above theorem is discussed in Theorem 2.17 of =-=[5]-=-, where the failure of the spectral mapping theorem for semigroups is demonstrated. Another type of example involving differential operators whose eigenvectors do not form a basis was presented in [8,... |

124 | Pseudospectra of linear operators
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Citation Context ...n has investigated a series of numerical examples using the concept of pseudospectrum (the contour plots of the resolvent norm), which provides a graphical demonstration of the degree of instability, =-=[20, 21]-=-. This concept is, however, not well adapted to the consideration of very large numbers of randomly generated matrices, for which one needs to produce a 1numerical measure of instability of the spect... |

93 |
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Citation Context ...elf-adjoint matrices. This problem does not occur for self-adjoint matrices because the variational theorem implies that the eigenvalues of such matrices do not change much under small perturbations, =-=[6]-=-. Our results indicate that nothing of a comparable nature is likely to be available in the non-self-adjoint case. We have also investigated a family of randomly generated non-self-adjoint bounded ope... |

59 |
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Citation Context ..., Trefethen has pointed out that the spectral behaviour of this type of operator 12is highly problematical.[22] This phenomenon has been investigated from several points of view over the last decade,=-=[2, 3, 7, 18, 19, 20, 21]-=- and among the conclusions is the warning that one cannot assume that a solution of a non-linear equation is stable simply because the eigenvalues of its linearization about the solution have negative... |

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Citation Context ...hat the vector (Bn+1,n, Bn,n, Bn,n+1) equals v if n ≥ 1 and equals w if n ≤ 0. The point of introducing this class of operators is that they are similar to operators whose spectrum is well understood.=-=[2, 3, 15]-=- Corollary 7 With probability 1 one has Spec(Aω) ⊇ ⋃ {σ(Bv,w) : v, w ∈ K} . Proof Every operator Bv,w lies in the limit class of A by Lemma 5. The significance of the above lemmas is best seen by appl... |

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Citation Context ...[5], where the failure of the spectral mapping theorem for semigroups is demonstrated. Another type of example involving differential operators whose eigenvectors do not form a basis was presented in =-=[8, 9, 10]-=-. It appears that such a situation is relatively common for non-self-adjoint operators in infinite dimensions. Since the eigenvectors do not form a basis for the Hilbert space, there is no reason to e... |

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(Show Context)
Citation Context ..., Trefethen has pointed out that the spectral behaviour of this type of operator 12is highly problematical.[22] This phenomenon has been investigated from several points of view over the last decade,=-=[2, 3, 7, 18, 19, 20, 21]-=- and among the conclusions is the warning that one cannot assume that a solution of a non-linear equation is stable simply because the eigenvalues of its linearization about the solution have negative... |

26 | the harmonic oscillator and complex resonances
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(Show Context)
Citation Context ...[5], where the failure of the spectral mapping theorem for semigroups is demonstrated. Another type of example involving differential operators whose eigenvectors do not form a basis was presented in =-=[8, 9, 10]-=-. It appears that such a situation is relatively common for non-self-adjoint operators in infinite dimensions. Since the eigenvectors do not form a basis for the Hilbert space, there is no reason to e... |

21 | Condition numbers of random triangular matrices
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Citation Context ... above theorem then it follows from the formula (A − z) −1 = ∑ Pi(λi − z) −1 that i ‖(A − z) −1 ‖ ≤ Nk dist(Spec(A), z) −1 Since the value of k frequently increases exponentially with the dimension N,=-=[20, 25]-=- one must expect pseudospectral information and that obtained from the instability index to be broadly equivalent. The point of the theorem is that if any spectral projection of A has very large norm,... |

19 |
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Citation Context ...the set of eigenvalues. Section 6 is devoted to spelling out the implications of our results for the non-self-adjoint Anderson model of HatanoNelson, which has been the focus of much recent attention.=-=[12, 13, 14, 17]-=- We find that the asymptotic behaviour of the eigenvalues as the volume increases does not describe the full spectrum of the infinite volume problem. The reason is that there are many approximate eige... |

13 |
Non-Hermitian localization and population biology, Phys
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(Show Context)
Citation Context ...the set of eigenvalues. Section 6 is devoted to spelling out the implications of our results for the non-self-adjoint Anderson model of HatanoNelson, which has been the focus of much recent attention.=-=[12, 13, 14, 17]-=- We find that the asymptotic behaviour of the eigenvalues as the volume increases does not describe the full spectrum of the infinite volume problem. The reason is that there are many approximate eige... |

13 |
Pseudospectra of Wiener-Hopf integral operators and constant coefficient differential operators
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(Show Context)
Citation Context ..., Trefethen has pointed out that the spectral behaviour of this type of operator 12is highly problematical.[22] This phenomenon has been investigated from several points of view over the last decade,=-=[2, 3, 7, 18, 19, 20, 21]-=- and among the conclusions is the warning that one cannot assume that a solution of a non-linear equation is stable simply because the eigenvalues of its linearization about the solution have negative... |

11 |
spectral behaviour of anharmonic oscillators
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(Show Context)
Citation Context ...[5], where the failure of the spectral mapping theorem for semigroups is demonstrated. Another type of example involving differential operators whose eigenvectors do not form a basis was presented in =-=[8, 9, 10]-=-. It appears that such a situation is relatively common for non-self-adjoint operators in infinite dimensions. Since the eigenvectors do not form a basis for the Hilbert space, there is no reason to e... |

11 |
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(Show Context)
Citation Context ...the set of eigenvalues. Section 6 is devoted to spelling out the implications of our results for the non-self-adjoint Anderson model of HatanoNelson, which has been the focus of much recent attention.=-=[12, 13, 14, 17]-=- We find that the asymptotic behaviour of the eigenvalues as the volume increases does not describe the full spectrum of the infinite volume problem. The reason is that there are many approximate eige... |

8 |
private communication
- Trefethen
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(Show Context)
Citation Context ...conditions are equally valid for a wide range of quasi-periodic boundary conditions. However, Trefethen has pointed out that the spectral behaviour of this type of operator 12is highly problematical.=-=[22]-=- This phenomenon has been investigated from several points of view over the last decade,[2, 3, 7, 18, 19, 20, 21] and among the conclusions is the warning that one cannot assume that a solution of a n... |

4 | On eigenfunction approximations for typical non-selfadjoint Schrödinger operators
- Aslanyan, Davies
- 1999
(Show Context)
Citation Context ... 0. The norm of a particular spectral projection is also called the condition number of the eigenvalue, and is known to measure how unstable the eigenvalue is under small perturbations of the matrix, =-=[1, 23, 24]-=-,[16, sect. 11.2]. We emphasize that if the norm of the spectral projection is very large the instability of the eigenvalue it intrinsic: it does not depend on the particular method of computing it. T... |

2 |
L 1 -properties of intrinsic Schrödinger operators
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- 1986
(Show Context)
Citation Context ...: the statements of our theorems only involve the set K rather than the probability measure µ. 8In order to prove some results about the spectra of operators in the family we introduce a notion from =-=[11]-=-. Given any bounded operator X on a Hilbert space H, we say that the operator Y lies in its limit class, Y ∈ C(X), if there exists a sequence Us of unitary operators on H such that U ∗ s XUs converges... |

1 |
H: The Algebraic Eigenvalue Problem
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- 1965
(Show Context)
Citation Context ... 0. The norm of a particular spectral projection is also called the condition number of the eigenvalue, and is known to measure how unstable the eigenvalue is under small perturbations of the matrix, =-=[1, 23, 24]-=-,[16, sect. 11.2]. We emphasize that if the norm of the spectral projection is very large the instability of the eigenvalue it intrinsic: it does not depend on the particular method of computing it. T... |