A General Scheme for Completion and Extension Problems

Abstract

This chapter presents an abstract scheme, called the band method, which allows one to deal with various positive and strictly contractive extension problems from one point of view. The theory developed here consists of three main elements. The first reduces the problem of finding a band extension to one of solving linear equations. The second is that all solutions of a positive extension problem are obtained via a linear fractional transformation of which the coefficients can be read off from a left and a right spectral factorization of the band extension. The third identifies the band extension in terms of an abstract maximum entropy principle. For strictly contractive extension problems this abstract approach has the same features, with triangular extensions in place of band extensions. This chapter consists of four sections. The first two develop the band method for positive extension problems, the third for strictly contractive extension problems, and the fourth presents the abstract maximum entropy principle. Applications appear in the next chapter.KeywordsBand StructureBanach AlgebraDiagonal EntryPositive SemidefiniteInvertible ElementThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.