Factorization of Matrix and Operator Functions: The State Space Method

Harm Bart · Israel Gohberg · Marinus A. Kaashoek · André C.M. Ran

dec. 2007 · Operator Theory: Advances and Applications Bog 178 · Springer Science & Business Media

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Thepresentbookdealswithvarioustypesoffactorizationproblemsformatrixand operator functions. The problems appear in di?erent areasof mathematics and its applications. A uni?ed approach to treat them is developed. The main theorems yield explicit necessaryand su?cient conditions for the factorizations to exist and explicit formulas for the corresponding factors. Stability of the factors relative to a small perturbation of the original function is also studied in this book. The unifying theory developed in the book is based on a geometric approach which has its origins in di?erent ?elds. A number of initial steps can be found in: (1) the theory of non-selfadjoint operators, where the study of invariant s- spaces of an operator is related to factorization of the characteristic matrix or operator function of the operator involved, (2) mathematical systems theory and electrical network theory, where a cascade decomposition of an input-output system or a network is related to a fact- ization of the associated transfer function, and (3) thefactorizationtheoryofmatrixpolynomialsintermsofinvariantsubspaces of a corresponding linearization. In all three cases a state space representation of the function to be factored is used, and the factors are expressed in state space form too. We call this approach the state space method. It hasa largenumber of applications.For instance, besides the areasreferred to above, Wiener-Hopf factorizations of some classes of symbols can also be treated by the state space method.