Commuting nonselfadjoint operators in Hilbert space : two independent studies / Moshe S. Livsic, Leonid L. Waksman

This book contans recent investigations to create an analogon of spectral theory of nonselfadjoint operators in the several commuting operators case.

The first part, written by M. S. Livšic, presents, with an extreme distinctness, the notion of “colligation” of a pair of bounded linear operators and its efficient frame: “the collective motions” of open systems. The fact that, for two operators with a commutative colligation, an input is a “collective” one iff it satisfies a wave equation (theorem 4) and the fact that, if the colligation is “commutative regular” the pair of operators satisfies an “algebraic equation” (theorem 8), is ingeniously interpreted. So, given a wave equation (like as the Schrödinger, Klein-Gordon or Dirac equations) it can be considered as an external manifestation of collective motions for which the dispersion law is the mentioned algebraic equation.

The second part of this book, in fact a monograph, is written by L. L. Waksman and is deeply related with Livšic’s ideas. It contains a theory of functional models (in the Nagy-Foias sense) of isometric and contractive representations of multiparameter semigroups and functional models of certain families of commuting dissipative operators. (Zentralblatt)