We study the pseudospectral theory of a variety of non-self-adjoint constant coefficient and variable coefficient differential operators, showing that the phenomenon of non-trivial pseudospectra is typical rather than exceptional. We prove that the pseudospectra provide more stable information about the operators under various limiting procedures than does the spectrum. AMS Subject Classification: 34L05, 35P05, 47A75, 49R99. Keywords: spectrum, pseudospectrum, norms of inverses, resolvent operators, differential operators. 1 Introduction It is well established that the spectrum of a self-adjoint operator is of crucial importance in understanding its action in various applied contexts. For highly non-self-adjoint operators, on the other hand, there is increasing evidence that the spectrum is often not very helpful, and that the pseudospectra are of more importance. We refer to [17, 18] for references to the increasing literature on this concept, and for a series of examples in which t...

We prove that there exist some Sturm-Liouville operators with square summable potentials such that the singular continuous component of the spectral measure lies on the positive halfline.

. We analyse the performance of five numerical methods for factoring a Laurent polynomial, which is positive on the unit circle, as the modulus squared of a real algebraic polynomial. It is found that there is a wide disparity between the methods, and all but one of the methods are significantly influenced by the variation in magnitude of the coefficients of the Laurent polynomial, by the closeness of the zeros of this polynomial to the unit circle, and by the spacing of these zeros. Keywords: spectral factorization, Toeplitz matrices, Euler-Frobenius polynomials, Daubechies wavelets. AMS subject classification: 12D05, 15A23. 1. Introduction Our recent interest in the asymptotic behaviour of the Gram-Schmidt iteration for orthonormalization of a large number of integer translates of a fixed function [11, 12] and also in techniques used for wavelet construction, cf. [7] and [20], has led us to experiment numerically with several existing algorithms to factor a Laurent polynomial, ass...

Many problems in physics and engineering with semi-infinite boundaries or interfaces are exactly solvable by the Wiener-Hopf technique. It has been used successfully in a multitude of different disciplines when the Wiener-Hopf functional equation contains a single scalar kernel. For complex boundary value problems, however, the procedure often leads to coupled equations which therefore have a kernel of matrix form. The key step in the technique is to decompose the kernel into a product of two functions, one analytic in an upper region of a complex (transform) plane and the other analytic in an overlapping lower half-plane. This is straightforward for scalar kernels but no method has yet been proposed for general matrices. In this article a new procedure is introduced whereby Pade approximants are employed to obtain an approximate but explicit noncommutative factorization of a matrix kernel. As well as being simple to apply, the use of approximants allows the accuracy of the factorization to be increased almost indefinitely. The method is demonstrated by way of example. Scattering of acoustic waves by a semi-infinite screen at the interface between two compressible media with different physical properties is examined in detail. Numerical evaluation of the approximate factorizations together with results on an upper bound of the absolute error reveal that convergence with increasing Pade number is extremely rapid. The procedure is computationally efficient and of simple analytic form and offers high accuracy (error < 10 -7 % typically). The method is applicable to a wide range of initial or boundary value problems.

this paper. We let (AP

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.

. This paper is concerned with Wiener-Hopf 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 ...

Abstract. This article traces several prominent trends in the development of the subject of holomorphic spaces, with emphasis on operator-theoretic aspects. The term “Holomorphic Spaces, ” the title of a program held at the Mathematical Sciences Research Institute in the fall semester of 1995, is short for “Spaces of Holomorphic Functions. ” It refers not so much to a branch of mathematics as to a common thread running through much of modern analysis—through functional analysis, operator theory, harmonic analysis, and, of course, complex analysis. This article will briefly outline the development of the subject from its origins in the early 1900’s to the present, with a bias toward operator-theoretic aspects, in keeping with the main emphasis of the MSRI program. I hope that the article will be accessible not only to workers in the field but to analysts in general. Origins The subject began with the thesis of P. Fatou [1906], a student of H. Lebesgue.

Abstract. These highly informal lecture notes aim at introducing and explaining several closely related problems on zeros of analytic functions defined by ordinary differential equations and systems of such equations. The main incentive for this study was its potential application to the tangential Hilbert 16th problem on zeros of complete Abelian integrals. The exposition consists mostly of examples illustrating various phenomena related to this problem. Sometimes these examples give an insight concerning the proofs, though the complete exposition of the latter is mostly relegated to separate expositions.

In this article various existence results for the LDU-factorization of semiinfinite and bi-infinite 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 non-banded Toeplitz matrices are considered. Extensive use is made of matrix polynomial theory. Results on the approximation by the LDU-factorizations of finite sections are discussed. The generalization of the results to the LDU-factorization of multi-index Toeplitz matrices is outlined. 1. Introduction Let A be a bi-infinite Toeplitz matrix A = (A i\Gammaj ) i;j2Z where Z is the set of integers and A h , h 2 Z, are k \Theta k matrices, so that A h is a scalar if k = 1 and a k \Theta k matrix if k ? 1. We consider the factorization A = LDU; (1.1) where L = (L i\Gammaj ) i;j2Z is a lower triangular matrix, U = (U i\Gammaj ) i;j2Z is an upper triangular matrix and D = (D i\Gammaj ) i;j2Z is a diagonal m...