Non-Self-Adjoint Operators and Pseudospectra

Transcription

1 Non-Self-Adjoint Operators and Pseudospectra E.B. Davies King s College London 1

2 The Relevance of Pseudospectra The theory of pseudospectra is a way of imposing some order on the chaos of non-self-adjoint (NSA) phenomena. It is relevant to: the study of complex resonances models involving optical potentials zeros of orthogonal polynomials on the unit circle stability problems for linear systems the spectral theory of Toeplitz operators, etc. The most important book on the subject is by L N Trefethen and M Embree, PUP

3 The Problem Spectral instability is the fact that Af λf < ε f for ε = (say) does not imply that λ is close to the spectrum of A. The existence of a non-zero vector f as above is equivalent to (zi A) 1 > ε 1. 3

4 The inequality (zi A) 1 dist(z, Spec(A)) 1. is an equality if A is a normal operator (i.e. AA = A A), but nothing which bears any resemblance to a reverse inequality holds in general. The graphics package EigTool enables one to compute the pseudospectral regions Spec ε (A) := {z : R(z, A) > ε 1 }. 4

5 Alternative Definitions It is easy to show that z Spec ε (A) if and only if for some f H, or Af zf < ε f for some B such that B < ε. z Spec(A + B) 5

6 The NSA harmonic oscillator H := P 2 + c 2 Q 2 acting in L 2 (R) has eigenvalues λ n := c(2n + 1) where n = 0, 1,... If c is complex then the norms of the spectral projections P n increase at an exponential rate as n. (EBD and Kuijlaars) 6

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8 The Jordan block (J 4 ) r,s := Using the explicit formula for (zi J n ) 1 one immediately obtains 0 (zi J n ) 1 1 = z n 1. 1 z 8

9 A = J n + δb where B = 1, n = 80 and δ =

10 Generalized Pseudospectra The standard theory of pseudospectra is a study of properties of the operator family zi A where A is a bounded or unbounded linear operator. One may consider polynomial operator pencils such as A(z) := n A r z r r=1 where A r are operators on some Banach space B. Examples are A zb and Az 2 + Bz + C. 10

11 By definition the spectrum of an operator family is the set Spec((A( )) := {z : A(z) is not invertible}. The pseudospectra are defined to be the sets Spec ε ((A( )) := S Spec(A( )) where S is the set of z for which there exists an approximate eigenvector f B satisfying A(z)f < ε f. 11

12 Truncation If L is the standard convection-diffusion operator (Lf)(u) := f (u) + f (u) acting in L 2 (R) then L is normal and its spectrum is the parabola {x + iy : x = y 2 }. If one truncates the operator to L 2 (0, a) and imposes DBC, one obtains an operator L a with spectrum {1/4 + π 2 n 2 /a 2 : n = 1, 2,...}. 12

13 Pseudospectra of the Truncation 13

14 Square Roots Consider (A) r,s := c 1/4 c 2/5 c 3/5 c 4/5 c Figure 2 plots A t against t for n = 100 and c = 0.6. Computing the square root of a NSA matrix may be a highly unstable procedure. The computations were carried out by diagonalizing A + B, where B is a random matrix of norm

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16 The above phenomenon is not explicable in terms of spectral theory. We have A t = e t log(a) and Spec(log(A)) = log(c) An explanation can be based on the pseudospectra of log(a). 16

17 Numerical Range The set Num(A) := { Af, f : f = 1}. is always convex, and satisfies Spec(A) Num(A) {z : z A }. There is a theorem stating that R(z, A) dist(z, Num(A)) 1. 17

18 Orthogonal Polynomials This section is a description of one part of my joint work with Barry Simon about the zeros of orthogonal polynomials. Given a measure µ with compact support in C one can apply Gram-Schmidt to 1, z, z 2, z 3,... regarded as lying in L 2 (C, µ). The zeros of these orthogonal polynomials all lie in the convex hull K of the support of µ. This is because the zeros are also the eigenvalues of a matrix whose numerical range is contained in K. 18

19 Theorem 1 Let A be an n n matrix and suppose that there exists a non-zero vector f such that Af zf ε f where z A. Then ( π dist(z, Spec(A)) ε cot 4n) and the constant on the RHS is optimal. 4nε π 19

20 The proof that the constant is optimal depends on identifying the worst possible matrix, which is M n =

21 This matrix has a number of special properties and is a discretization of the standard Volterra integral operator (V f)(x) := x 0 f(s) ds acting on L 2 (0, 1). The proof of the theorem relies upon the fact that z is close to the boundary of the numerical range of A. 21

22 A More General Version Theorem 2 Let A be an n n matrix and suppose that z C is not in the interior of the numerical range of A. If there exists a non-zero vector f such that Af zf ε f then ( π dist(z, Spec(A)) ε cot. 4n) 22