Matrix and operator extensions
Hugo J. Woerdeman · Centrum Wiskunde & Informatica (CWI), the national research institute for mathematics and computer science in the Netherlands · 1989
In our contractive extension problem the operators again act on X = Y = c;n or on X = Y = 1 2 .But now the given part is of triangular form and consists of all diagonals below a given diagonal, the q -th say.The extensions are required to have operator norm tive extension problem may be rephrased as an extension problem concerning (matrix-or operator-valued) functions on the unit circle.In what follows we treat the above problems in more detail, and we describe some of our main results. Positive extension problems.Consider the following problem.Let B ij be given In a paper by J.A. Ball and I. Gohberg [5] a finite dimensional version of the shift invariant subspace approach of J.A. Ball and J.W. Helton [6] was used to derive a full parametrization of the set of all solutions via a linear fractional representation.The coefficients in this linear fractional map are determined using a theorem of Beurling-Lax type.In this book we shall present two other methods to obtain such a linear fractional representation and, moreover, we shall give explicit formulas for the coefficients in this linear fractional map in terms of the given data.The following two theorems are among our main results.THEOREM 0.1.Let B Ii = B ; 1 be given matrices for I j -i I ~ q ( ~ nl), and assume that condition (0.2) is satisfied.For 0 < j-i ~ q define the matrix Zu by where {j ii ,'Y iJ and r ii are given via the partitioning Put and let a 0 , ••• , Aq be defined recursively by Put IJ i,j=I IJ iJ=I ' j ; § i ; § {3(j); , elsewhere; B = B2 B1 Bo ... . . . .such that Bi = Ai, IJ I ~ m, and the symbol of B is in the Wiener algebra.In this book we illustrate the band method only on the above mentioned examples.