## Eigenvalue estimates for non-normal matrices and the zeros of random orthogonal polynomials on the unit circle

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Venue: | J. Approx. Theory |

Citations: | 6 - 3 self |

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@ARTICLE{Davies_eigenvalueestimates,

author = {E. B. Davies and Barry Simon},

title = {Eigenvalue estimates for non-normal matrices and the zeros of random orthogonal polynomials on the unit circle},

journal = {J. Approx. Theory},

year = {},

volume = {141},

pages = {189--213}

}

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### Abstract

Abstract. We prove that for any n × n matrix, A, and z with |z | ≥ �A�, we have that �(z − A) −1 � ≤ cot ( π 4n)dist(z, spec(A))−1. We apply this result to the study of random orthogonal polynomials on the unit circle. 1.

### Citations

158 |
Harmonic Analysis of Operators on Hilbert Space
- Sz-Nagy, Foias
- 1970
(Show Context)
Citation Context ...(A−1)y�)2/3 and using 1 6n + ≤ 3n, we obtain 2 π (6.23). � E. Von Neumann’s theorem. Lemma 3.1 is a special case of a theorem of von Neumann. The now standard proof of this result uses Nagy dilations =-=[23]-=-; we have found a simple alternative that relies on Lemma 6.5. For any A, with �A� < 1 and A = U|A|, and U unitary, there exists an operator-valued function, g, analytic in a neighborhood of D so that... |

151 |
Orthogonal polynomials on the unit circle. Part 1: Classical theory and Part 2: Spectral Theory
- Simon
- 2005
(Show Context)
Citation Context ...−1 � −1 (1.10) (1.11) We were motivated by seeking a replacement of (1.9) in a case where An is non-normal. Indeed, we had a specific situation of orthogonal polynomials on the unit circle (OPUC; see =-=[17, 18]-=-) where one has a sequence zn ∈ ∂D = {z | |z| = 1} and corresponding unit trial vectors, yn, so that �(An − zn)yn� ≤ C1e −C2n (1.12) for all n with C2 > 0. We would like to conclude that Φn(z) has zer... |

114 | One-Parameter Semigroups - Davies - 1980 |

113 | Spectra and Pseudospectra, The Behavior of Nonnormal Matrices and Operators
- Trefethen, Embree
- 2005
(Show Context)
Citation Context ...orderline between the dimension-independent bound (1.27) for |z| > �A� and the exponential growth that may happen if |z| < �A�, essentially the phenomenon of pseudospectra which is well documented in =-=[24]-=-; see also [15]. The structure of this paper is as follows. In Section 2, we will prove Theorem 4, the most significant result in this paper since it implies c(n) < ∞ and, indeed, with no effort that ... |

102 |
B.: Analysis of Toeplitz operators
- Böttcher, Silbermann
- 1990
(Show Context)
Citation Context ...λℓ| −2 (1 − |λk| 2 )(1 − |λℓ| 2 )] 1/2 ≤ 2[|1 − λk| −1 |1 − λℓ| −1 ] 1/2 ≤ 2[dist(1, spec(A))] −1 (2.9) by (2.2). �s8 E. B. DAVIES AND B. SIMON 3. Upper Triangular Toeplitz Matrices A Toeplitz matrix =-=[1]-=- is one that is constant along diagonals, that is, Ajk is a function of j − k. An n × n upper triangular Toeplitz matrix (UTTM) is thus of the form ⎛ ⎞ a0 a1 a2 . . . an−1 ⎜ 0 a0 a1 . . . an−2⎟ ⎜ ⎝ . ... |

88 | Orthogonal Polynomials on the Unit Circle, Part 2 - Simon - 2005 |

86 |
Über Potenzreihen die im Innern des Einheitskreises beschränkt sind. Journal für die Reine und Angewandte Mathematik
- Schur
- 1917
(Show Context)
Citation Context ...ecause the operators, An(a), of Theorem 3 will be of this form. In this section, after recalling the basics of UTTM, we will prove Theorem 3. Then we will state some results, essentially due to Schur =-=[16]-=-, on the norms of UTTM that we will need in Section 5 in one calculation of the norm of Mn. Given any function, f, which is analytic near zero, we write Tn(f) for the matrix in (3.1) if f(z) = a0 + a1... |

83 |
Problems and Theorems in Analysis I
- PÓLYA, SZEGÖ
- 1972
(Show Context)
Citation Context ...s theorem. � Our second proof relies on the following known result (see Milovanić et al. [5], page 272, and references therein; this result is called the Eneström-Kakeya theorem; see also Pólya-Szegő =-=[14]-=-, problem 22 on pp. 107 and 301, who also mention Hurwitz):s16 E. B. DAVIES AND B. SIMON Lemma 5.3. Suppose 0 < a0 < a1 < · · · < an Then P(z) = a0 + a1z + · · · + anz n has all its zeros in D. Theore... |

65 |
Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle, Linear Algebra Appl
- Cantero, Moral, et al.
- 1987
(Show Context)
Citation Context ... 2 2π where |I| is the dθ measure of I. For the next theorem, we need the fact that there is an explicit realization of An and the associated rank one perturbations as n × n complex CMV matrices (see =-=[2, 17, 18, 19]-=-), Cn, whose eigenvalues are , so that the z n j , and ˜ C (βn) n whose eigenvalues are the ˜z n j �(Cn − C (βn) n )ϕ� ≤ |ϕn−1| + |ϕn| (7.8) The next theorem uses the components so (7.8) holds. Theore... |

61 |
Hankel Operators and Their Applications
- Peller
(Show Context)
Citation Context ...n−1 is the start of the Taylor series of a Schur function if and only if the matrix A of (3.1) obeys A ∗ A ≤ 1. This result is intimately connected to Nehari’s theorem on the norm of Hankel operators =-=[8, 13]-=-; see Partington [12]. 3. This is classical; see [1, 10, 13]. To state the last result of this section, we need a definition: Definition. A Blaschke factor is a function on D of the form z − w f(z, w)... |

53 |
Local fluctuation of the spectrum of a multidimensional Anderson tight binding model
- Minami
- 1996
(Show Context)
Citation Context ...end heavily on earlier results of Stoiciu [20, 21], who studied a closely related problem (see below). In turn, Stoiciu relied, in part, on earlier work on eigenvalues of random Schrödinger operators =-=[7, 6]-=-. We will prove the following three theorems: Theorem 7.1. Let 0 < ρ < 1. Let k ∈ {1, 2, . . . }. Then for a.e. ω in the ρ-model, lim sup n→∞ #{j | |z (n) j (ω)| < 1 − n −k } [log(n)] 2 < ∞ (7.3) Thus... |

45 |
On bounded bilinear forms
- Nehari
- 1957
(Show Context)
Citation Context ...n−1 is the start of the Taylor series of a Schur function if and only if the matrix A of (3.1) obeys A ∗ A ≤ 1. This result is intimately connected to Nehari’s theorem on the norm of Hankel operators =-=[8, 13]-=-; see Partington [12]. 3. This is classical; see [1, 10, 13]. To state the last result of this section, we need a definition: Definition. A Blaschke factor is a function on D of the form z − w f(z, w)... |

34 |
Bounds for iterates, inverses, spectral variation and field of values of non–normal matrices
- Henrici
- 1962
(Show Context)
Citation Context ... n position. Thus, as is well known, �(An − z) −1 � for general n × n matrices An and general z cannot be bounded by better than dist(z, spec(An)) −n . Indeed, the existence of such bounds by Henrici =-=[4]-=- is part of an extensive literature on general variational bounds on eigenvalues. Translated to a variational bound, this would give dist(zn, {zeros of Φn}) ≤ C�(An − zn)y� 1/n , which would not give ... |

34 | OPUC on one foot
- Simon
(Show Context)
Citation Context ...g the self-adjoint part (see also Nikolski [10]). 7. Zeros of Random OPUC In this section, we apply Theorem 1 to obtain results on certain OPUC. We begin by recalling the recursion relations for OPUC =-=[17, 18, 19]-=-. For each non-trivial probability measure, dµ, on ∂D, there is a sequence of complex numbers, {αn(dµ)} ∞ n=0 , called Verblunsky coefficients so that where Φn+1(z) = zΦn(z) − ¯αnΦ ∗ n (z) (7.1) Φ ∗ n... |

25 |
An Introduction to Hankel Operators
- Partington
- 1989
(Show Context)
Citation Context ...Taylor series of a Schur function if and only if the matrix A of (3.1) obeys A ∗ A ≤ 1. This result is intimately connected to Nehari’s theorem on the norm of Hankel operators [8, 13]; see Partington =-=[12]-=-. 3. This is classical; see [1, 10, 13]. To state the last result of this section, we need a definition: Definition. A Blaschke factor is a function on D of the form z − w f(z, w) = 1 − wz (3.27) wher... |

22 |
The local structure of the spectrum of the one-dimensional Schrödinger operator
- Molchanov
- 1981
(Show Context)
Citation Context ...end heavily on earlier results of Stoiciu [20, 21], who studied a closely related problem (see below). In turn, Stoiciu relied, in part, on earlier work on eigenvalues of random Schrödinger operators =-=[7, 6]-=-. We will prove the following three theorems: Theorem 7.1. Let 0 < ρ < 1. Let k ∈ {1, 2, . . . }. Then for a.e. ω in the ρ-model, lim sup n→∞ #{j | |z (n) j (ω)| < 1 − n −k } [log(n)] 2 < ∞ (7.3) Thus... |

22 |
Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes
- Neumann
- 1951
(Show Context)
Citation Context ... z + |A| g(z) = U 1 + z|A| The factor in [. . .] is unitary if z = e iθ , since (e iθ + |A|) ∗ (e iθ + |A|) = 1 + A ∗ A + 2 cosθ|A| (6.32) = (1 + e iθ |A|) ∗ (1 + e iθ |A|) � Theorem 6.6 (von Neumann =-=[25]-=-). Let f : D → D. If �A� < 1, define f(A) by ∞� f(z) = anz n ∞� f(A) ≡ anA n (6.33) Then n=0 n=0 �f(A)� ≤ 1 (6.34) Proof of von Neumann’s theorem, given the lemma. Suppose first that A obeys the hypot... |

17 |
Systems: an easy
- Nikolski, Function
- 2002
(Show Context)
Citation Context ...n if and only if the matrix A of (3.1) obeys A ∗ A ≤ 1. This result is intimately connected to Nehari’s theorem on the norm of Hankel operators [8, 13]; see Partington [12]. 3. This is classical; see =-=[1, 10, 13]-=-. To state the last result of this section, we need a definition: Definition. A Blaschke factor is a function on D of the form z − w f(z, w) = 1 − wz (3.27) where w ∈ D. A (finite) Blaschke product is... |

14 | The statistical distribution of the zeros of random paraorthogonal polynomials on the unit circle
- Stoiciu
(Show Context)
Citation Context ...s College London. He would like to thank A. N. Pressley and E. B. Davies for the hospitality of King’s College, and the London Mathematical Society for partial support. The calculations of M. Stoiciu =-=[20, 21]-=- were an inspiration for our pursuing the estimate we found. We appreciate useful correspondence/discussions with M. Haase, N. Higham, R. Nagel, N. K. Nikolski, V. Totik, and L. N. Trefethen. 2. The K... |

11 |
A Concise Introduction to the Theory of Integration
- Stroock
- 1998
(Show Context)
Citation Context ... O((nn −4 ) 2 ) = O(n −6 ). The number of intervals at order n is O(n 4 ). Since � ∞ n=1 n4 n −6 < ∞, the sum of the probabilities of two zeros in an interval is summable. By the Borel-Cantelli lemma =-=[22]-=- for a.e. ω, only finitely many intervals have two zeros. Hence, for large n, (7.17) holds. � Proof of Theorem 7.1. Obviously, if (7.3) holds for some k, it holds for all smaller k, so we will prove i... |

7 |
Condition Numbers of Large Matrices and Analytic Capacities
- Nikolski
(Show Context)
Citation Context ... get information from (1.12). The key realization is that zn and �An� are not general. Indeed, |zn| = �An� = 1 (1.16) It is not a new result that a linear bound holds in the generality we discuss. In =-=[11]-=-, Nikolski presents a general method for estimating norms of inverses in terms of minimal polynomials (see the proof of Lemma 3.2 of [11]) that is related to our argument in Subsection 6A. His ideas y... |

4 |
The distinguished boundary of the unit operator ball
- Nelson
(Show Context)
Citation Context ...This is automatic in the finite-dimensional case and also if dim(H) = ∞ for A ⊕ 0 since then both spaces are infinite-dimensional.s22 E. B. DAVIES AND B. SIMON 2. This proof is close to one of Nelson =-=[9]-=- who also uses the maximum principle and polar decomposition, but uses a different method for interpolating the self-adjoint part (see also Nikolski [10]). 7. Zeros of Random OPUC In this section, we ... |