Hilbert and Krein space operators: extension problems and functional models

Informations projet

N° de convention de subvention: INTAS-93-0249

Projet clôturé

Date de début 1 Novembre 1994

Date de fin 31 Decembre 1995

Financé au titre de

• International Association for the promotion of cooperation with scientists from the independent states of the former Soviet Union (INTAS), 1993-

Coût total

Aucune donnée

Contribution de l’UE

€ 40 000,00

Coordonné par

Vrije Universiteit Amsterdam
Netherlands

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Hilbert and Krein space operators: extension problems and functional models

Linear operators are used to describe a great variety of physical phenomena. Depending on the kind of problem, the operators may appear as finite or infinite matrices, as integral operators or as ordinary and partial differential operators. Two important methods exist to study operators in Hilbert and Krein spaces, each of which has a long and outstanding history. The first method consists of extending a given operator to one acting on a larger space and with simpler properties. The second approach is based on the construction of canonical models for operators. Functional models provide powerful tools to study non-selfadjoint Hilbert space operators that are close to unitary or selfadjoint ones. Extension of operators originates in the theory of symmetric operators. Both approaches enable the study of operators by methods of complex function theory and, vice versa, the study of complex function theory problems by methods of operator theory. The purpose of the project is to employ and further develop these two methods. The objectives are the following: to study the classification of operators under similarity and quasi-similarity (in particular to describe the spectra of selfadjoint extensions of symmetric operators and symmetric linear relations and to study the analytic matrix and operator functions that are involved); to investigate the functional models determined by transfer functions of dynamical systems; to develop a non-stationary version of the model theory; to develop scattering theory for non-selfadjoint operators in terms of functional models and for pairs of selfadjoint extensions of a single symmetric operator (including trace formula and spectral shift function); and to study inverse problems of mathematical physics.