For ? > 0 and a bounded linear operator T acting on some Hilbert space, the ?-pseudospectrum of is ??(T) = {z C : ||(zI-T)-1|| ?-1}. This note provides characterization those operators satisfying ?(T) + B(0,?) for all 0. Here |z| < ?}. In particular, such finite dimensional spaces must be normal.

We show that it is possible to compute spectra and pseudospectra of linear operators on separable Hilbert spaces given their matrix elements. The core in the theory is pseudospectral analysis and in particular the n-pseudospectrum and the residual pseudospecrum. We also introduce a new classification tool for spectral problems, namely, the Solvability Complexity Index. This index is an indicato...

A. Ferté, J. Peloton, J. Grain, and R. Stompor Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, United Kingdom Université Paris-Sud 11, Institut d’Astrophysique Spatiale, UMR8617, Orsay F-91405, France CNRS, Orsay F-91405, France AstroParticule et Cosmologie, Université Paris Diderot, CNRS/IN2P3, CEA/Irfu, Observatoire de Paris, Sorbonne Pa...

The famous two weights problem consists in characterising all possible pairs of weights such that the Hardy projection is bounded between the corresponding weighted L spaces. Koosis’ theorem of 1980 gives a way to construct a certain class of pairs of weights. We show that Koosis’ theorem is closely related to (in fact, is a direct consequence of) a spectral perturbation model suggested by de B...

One of the conditions in the Kreiss matrix theorem involves the resolvent of the matrices A under consideration. This so-called resolvent condition is known to imply, for all n ≥ 1, the upper bounds ‖An‖ ≤ eK(N + 1) and ‖An‖ ≤ eK(n + 1). Here ‖ · ‖ is the spectral norm, K is the constant occurring in the resolvent condition, and the order of A is equal to N + 1 ≥ 1. It is a long-standing proble...

We prove a spectral perturbation theorem for rank-one updated matrices of special structure. Two applications of the result are given to illustrate the usefulness of the theorem. One is for the spectrum of the Google matrix and the other is for the algebraic simplicity of the maximal eigenvalue of a positive matrix. c © 2007 Elsevier Ltd. All rights reserved.

In this report, we implement a method for computing L∞-norms for descriptor systems using structured iterative eigensolvers. In particular, the algorithm computes some desired imaginary eigenvalues of an even matrix pencil and uses them to determine an upper and lower bound to the L∞-norm. We finally compare our method to a previously developed algorithm using structured pseudospectra. Numerica...

We study differential equations that lead to extremal points in symplectic pseudospectra. In a two-level approach, where on the inner level we compute extremizers of the symplectic ε-pseudospectrum for a given ε and on the outer level we optimize over ε, this is used to solve symplectic matrix nearness problems such as the following: For a symplectic matrix with eigenvalues of unit modulus, we ...

We will investigate the condition number, eigenvalue perturbations and pseudospectrum of Toeplitz matrices under structured perturbations. Sometimes we will see few changes, sometimes, although provably rare, dramatic differences in the structured and unstructured view. Connections to minimization problems concerning polynomials are shown. One result is that the structured distance to the neare...

This paper studies convergence properties of the block GMRES algorithm when applied to nonsymmetric systems with multiple right-hand sides. A convergence theory is developed based on a representation of the method using matrix-valued polynomials. Relations between the roots of the residual polynomial for block GMHES and the matrix &-pseudospectrum are derived, and illustrated with numerical exp...