Bøger af Marinus A. Kaashoek

This monograph presents necessary and sufficient conditions for completeness of the linear span of eigenvectors and generalized eigenvectors of operators that admit a characteristic matrix function in a Banach space setting. Classical conditions for completeness based on the theory of entire functions are further developed for this specific class of operators. The classes of bounded operators that are investigated include trace class and Hilbert-Schmidt operators, finite rank perturbations of Volterra operators, infinite Leslie operators, discrete semi-separable operators, integral operators with semi-separable kernels, and period maps corresponding to delay differential equations. The classes of unbounded operators that are investigated appear in a natural way in the study of infinite dimensional dynamical systems such as mixed type functional differential equations, age-dependent population dynamics, and in the analysis of the Markov semigroup connected to the recently introduced zig-zag process.

Thisvolumecontainsaselectionofpapersinmodernoperatortheoryanditsapp- cations. Most of them are directly related to lectures presented at the Fourteenth International Workshop on Operator Theory and its Applications (IWOTA 2003) held at the University of Cagliari, Italy, in the period of June 24-27, 2003. The workshop, which was attended by 108 mathematicians - including a number of PhD and postdoctoral students - from 22 countries, presented eight special sessions on 1) control theory, 2) interpolation theory, 3) inverse scattering, 4) numerical estimates for operators, 5) numerical treatment of integral equations, 6) pseudodi?erential operators, 7) realizations and transformations of analytic functions and inde?nite inner product spaces, and 8) structured matrices. The program consisted of 19 plenary lectures of 45 minutes and 78 lectures of 30 minutes in four parallel sessions. The present volume re?ects the wide range and rich variety of topics p- sentedanddiscussedattheworkshop,bothwithinandoutsidethespecialsessions. The papers deal with inverse scattering, numerical ranges, pseudodi?erential - erators, numerical analysis, interpolation theory, multidimensional system theory, inde?nite inner products, spectral factorization, and stationary processes. SinceintheperiodthattheproceedingsofIWOTA2003werebeingprepared, Israel Gohberg, the president of the IWOTA steering committee, reached the age of 75, we decided to dedicate these proceedings to Israel Gohberg on the occasion of his 75th birthday. All of the authors of these proceedingshavejoined the editors and dedicated their papers to him as well.

The present book deals with canonical factorization of matrix and operator functions that appear in state space form or that can be transformed into such a form. The main results are all expressed explicitly in terms of matrices or operators, which are parameters of the state space representation.