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Approximation of the Ε-Pseudospectra

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Approximation of the Ε-Pseudospectra APPROXIMATION OF THE ε-PSEUDOSPECTRA OF TOEPLITZ OPERATORS A Thesis Presented to the Faculty of California State Polytechnic University, Pomona In Partial Fulfillment Of the Requirements for the Degree Master of Science In Mathematics By Christine Kchech 2015 SIGNATURE PAGE THESIS: APPROXIMATION OF THE ε-PSEUDOSPECTRA OF TOEPLITZ OPERATORS AUTHOR: Christine Kchech DATE SUBMITTED: Fall 2015 Department of Mathematics and Statistics Dr. Arlo Caine Thesis Committee Chair Mathematics & Statistics Dr. Hubertus Von Bremen Mathematics & Statistics Dr. Randall Swift Mathematics & Statistics ii ACKNOWLEDGMENTS Thank you God for everything. I would like to thank my thesis adviser Dr. Caine for mentoring me throughout this process and making sure I did not give up. This paper would not have been possible if it was not for him. I would like to thank Dr. Von Bremen for always pushing me to accomplish more in my academic life as well as personal life. I would like to thank Dr. Swift for always ensuring me that I am doing well in my classes as well introducing me to Differential Equations which I grew to love. I would like to thank Dr. Rosin for not just being my professor but also my mentor. Thank you to my parents and sister for always supporting me. I would like to thank the Mathematics and Statistics department for their support. I would also like to acknowledge the friends that I have made throughout the program for making my stay at Cal Poly Pomona worthwhile. Lastly, I would like to thank Erik Augustine for helping me with MATLAB. The block code would not have been possible without you. iii ABSTRACT This paper investigates the relationship between the spectrum of a Toeplitz operator and the spectra of its approximating Toeplitz matrices. The main tool in this investi­ gation is the concept of ε-pseudo-eigenvalues introduced by Reichel and Trefethen. The four equivalent conditions defining ε-pseudo-eigenvalues lead to four different numeri­ cal computation techniques for approximating the ε-pseudospectrum. The first of these routines generates pseudo-eigenvalues with a non-uniform statistical distribution and we investigate these distributions in varies examples, attempting to determine whether the ε-pseudo-eigenvalues of a large Toeplitz matrix AN concentrate near the eigenvalues of AN. iv Contents List of Figures ix 1 Introduction 1 1.1 Inspiration ... .................................. 1 1.2 Toeplitz Matrices and Operators ........................ 2 1.3 The Toeplitz Theorem .............................. 7 1.4 Fourier Analysis and Toeplitz Operators ................... 10 1.5 Trigonometric Moment Problem ........................ 13 2 Pseudo-Eigenvalues 14 2.1 Definition of the Pseudospectrum ....................... 14 2.2 Approximating the Pseudospectrum ... ................... 18 2.2.1 Perturbation code ............................ 18 2.2.2 Unitary code ............................... 23 2.2.3 Resolvent code ............................. 26 2.2.4 Singular Value Decomposition code ................. 29 2.2.5 Comparing Theoretical vs. Numerical Aspects ........... 32 2.3 Reichel-Trefethen Theorems for Toeplitz operators ............. 33 v 3 Results 37 3.0.1 Triangular A ............................... 37 3.0.2 Triangular B ............................... 40 3.0.3 Triangular C ............................... 42 3.1 Results for a General Toeplitz Matrix AN ................... 44 3.2 Conclusion .. .................................. 47 3.3 Further Studies .................................. 48 Bibiliography 55 vi List of Tables 2.1 Run times of the perturbation code for different N and ε values for 2 Toeplitz matrix AN with symbol fA(z)=z +z using perturbation code. 20 2.2 Run times for different N and ε values for Toeplitz matrices A with sym­ 2 bol fA(z)=z +z using unitary code. ...................... 25 2.3 Run times for different N and ε values for Toeplitz matrices A with sym­ 2 bol fA(z)=z +z using the resolvent code. .................. 28 2.4 Run times for different N and ε values for Toeplitz matrices A with sym­ 2 bol fA(z)=z +z using singular value decomposition code. ....... 31 vii List of Figures 1.1 Regions and winding numbers for the Toeplitz matrix A with symbol −1 2 3 fA(z)=2iz +z +0.7z . ............................ 10 2.1 Approximation of Λε (A) generated by the perturbation code for the 100× 2 100 Toeplitz matrix AN with symbol fA(z)=z +z . ............. 20 2.2 Approximation of Λε (A) for 100 × 100 Toeplitz matrix A with symbol 2 fA(z)=z +z using modified perturbation code. ............... 22 2.3 Approximation of Λε (A) generated by the unitary code for the 100×100 2 Toeplitz matrix A with symbol fA(z)=z +z and ε = 2. ........... 26 2.4 Approximation of Λε (A) generated by the resolvent code for the 100 × 2 100 Toeplitz matrix A with symbol fA(z)=z +z . .............. 29 2.5 Approximation of Λε (A) generated by the singular value decomposition 2 code 100 ×100 Toeplitz matrix A with symbol fA(z)=z +z ......... 32 2.6 Approximation of Λ(A) as N increases and ε decreases. .......... 35 2.7 Approximation of Λε (A) with ε = 0.1 and 1000×1000 Toeplitz matrix A 2 with symbol fA(z)=z +z . ........................... 36 2 3.1 Approximation of Λε (A) for Toeplitz matrix A with symbol fA(z)=z+z using different N and ε values. ......................... 39 viii Λ ( ) ( )= 1+z 3.2 Approximation of ε B for Toeplitz matrix B with symbol fB z 1−z using different N and ε values. ... ...................... 41 3.3 Approximation of Λε (C) for Toeplitz matrix C with symbol fC(z)=z using different N and ε values. ......................... 43 3.4 Approximation of Λε (A) for Toeplitz matrix A with symbol fA(z)= 2iz−1 +z2 +0.7z3 using different N and ε values. ............... 46 3.5 Approximation of Λε (A) and Λ(A) for Toeplitz matrix A with sym­ −1 2 3 bol fA(z)=2iz + z + 0.7z using N = 500 and ε = 0.002 values where Λε (AN ) in blue and Λ(AN ) in red. ....................... 47 3.6 Approximation of Λε (A) for block Toeplitz matrix A using different N and ε values. ................................... 53 ix Chapter 1 Introduction The spectrum of a finite dimensional matrix can be computed numerically but the spectrum of an operator cannot. We will look at the spectrum of both matrices and operators as well as the connection between Fourier series and Toeplitz operators. 1.1 Inspiration Toeplitz operators were introduced in 1910 by Toeplitz [11]. Toeplitz was studying the spectral theory of operators on infinite dimensional spaces. Toeplitz matrices and op­ erators have been used in many fields, such as Physics, probability theory, and function theory because their relationship to Fourier analysis. Since it is impossible to numer­ ically find the eigenvalues of Toeplitz operators, the idea of ε-pseudo-eigenvalues was introduced. The concept of pseudo-eigenvalues has been reinvented at least five times [11]. The study of ε-pseudospectra of Toeplitz matrices helps to study the spectra of Toeplitz operators, and vice versa. These connections will be explained in chapters 1 and 2. 1 1.2 Toeplitz Matrices and Operators Definition 1.2.1. A Toeplitz matrix with coefficients (a1−N,...,a−1,a0,a1,...,aN−1) in C is a N ×N matrix of the form: ⎛ ⎞ a a ... a − a − ⎜ 0 1 N 2 N 1⎟ ⎜ ⎟ ⎜ ⎟ ⎜ a− a ⋱ ⋮ ⋮ ⎟ ⎜ 1 0 ⎟ ⎜ ⎟ = ⎜ ⎟. AN ⎜ a−2 a−1 ⋱ a1 a2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⋮ ⋮ ⋱ a0 a1 ⎟ ⎜ ⎟ ⎝ ⎠ a1−N a2−N ... a−1 a0 N−1 ( )= ∑ k The symbol of a Toeplitz matrix AN is a complex function f defined by fAN z akz . k=1−N Recall that a linear operator on CN is a function L ∶ CN → CN such that N 1. L(c1 +c2)=L(c1)+L(c2), for all c1,c2 ∈ C 2. L(kc)=kL(c) for all c ∈ CN and all k ∈ C. Since matrix multiplication distributes over vector addition and commutes with scalar multiplication, every N × N Toeplitz matrix defines a linear operator on CN. Next, we will define a Toeplitz operator, which is an infinite Toeplitz matrix acting on a space of ∞ 2 sequences (an) = in C. The space l (N) is the vector space of all infinite sequences n 0 √ ( , ,...) ∞ ∣ ∣2 ∥( , ,...)∥ = ∞ ∣ ∣2 a0 a1 such that ∑n=0 an converges, with norm a0 a1 2 ∑n=0 an . ( )∞ C Definition 1.2.2. A Toeplitz operator A with coefficients ak k=−∞ in is a linear oper­ ator on l2(N) given by multiplication by the infinite matrix ⎛ a a a ...⎞ ⎜ 0 1 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜a− a a ⋱ ⎟ = ⎜ 1 0 1 ⎟. A ⎜ ⎟ ⎜ ⎟ ⎜a−2 a−1 a0 ⋱ ⎟ ⎜ ⎟ ⎝ ⋮ ⋱ ⋱ ⋱ ⎠ 2 ∞ k The symbol of the Toeplitz operator is the complex function fA(z)= ∑ akz . Note that k=−∞ the sequence of coefficients of such a matrix must decay in order for multiplication to 2 k preserve l (N) and for the series ∑akz to define a function. The symbol f (z) belongs in the Wiener class if the sequence of coefficients is absolutely summable. We write S1 for the unit circle in C and L2(S1) for the Hilbert space of functions on 1 1 ( ) ( ) 2( 1) S with the inner product 2πi ∫S1 f z g z dz. Then L S + denotes the subspace of the boundary values of functions analytic in the open unit disk D. ( )= ∞ n Lemma 1.2.3. If the symbol fA z ∑n=0 anz belongs to the Wiener class then A is bounded on l2(N). Proof. We will provide some idea on the proof but refer to Widom [9] for the full argu­ ( )= ∞ n , , 2 ,... ment. Suppose A is a Toeplitz operator with symbol fA z ∑n=0 anz . Since 1 z z 2 1 2 is an orthogonal basis for L (S )+ then multiplication by A on l (N) is equivalent to 2( 1) ∈ 2(N) ( )= ∞ n multiplication by the symbol fA on L S +.
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