Catálogo de publicaciones - libros

We review our recent work on the Schur transformation for scalar generalized Schur and Nevanlinna functions. The Schur transformation is defined for these classes of functions in several situations, and it is used to solve corresponding basic interpolation problems and problems of factorization of rational -unitary matrix functions into elementary factors. A key role is played by the theory of reproducing kernel Pontryagin spaces and linear relations in these spaces.

The main goal of this paper is to study the truncated matricial moment problem on a finite closed interval in the case of an odd number of prescribed moments by using of the FMI method of V.P. Potapov. The solvability of this problem is characterized by the fact that two block Hankel matrices built from the data of the problem are nonnegative Hermitian (Theorem 1.3). An essential step to solve the problem under consideration is to derive an effective coupling identity between both block Hankel matrices (Proposition 2.5). In the case that these Hankel matrices are both positive Hermitian we parametrize the set of solutions via a linear fractional transformation the generating matrix-valued function of which is a matrix polynomial whereas the set of parameters consists of distinguished pairs of meromorphic matrix-valued functions.

In this paper we construct a class of homogeneous Hilbert modules over the disc algebra as quotients of certain natural modules over the function algebra . These quotient modules are described using the jet construction for Hilbert modules. We show that the quotient modules obtained this way, belong to the class B() and that they are mutually inequivalent, irreducible and homogeneous.

Canonical forms are given for pairs of quaternionic matrices, or equivalently matrix pencils, with various symmetry properties, under strict equivalence and symmetry respecting congruence. Symmetry properties are induced by involutory antiautomorphisms of the quaternions which are different from the quaternionic conjugation. Some applications are developed, in particular, canonical forms for quaternionic matrices that are symmetric or skewsymmetric with respect to symmetric or skewsymmetric quaternion-valued inner products. Another application concerns joint numerical cones of pairs of skewsymmetric quaternionic matrices.

‘Hierarchical Semi-separable’ matrices (HSS matrices) form an important class of structured matrices for which matrix transformation algorithms that are linear in the number of equations (and a function of other structural parameters) can be given. In particular, a system of linear equations = can be solved with linear complexity in the size of the matrix, the overall complexity being linearly dependent on the defining data. Also, LU and ULV factorization can be executed ‘efficiently’, meaning with a complexity linear in the size of the matrix. This paper gives a survey of the main results, including a proof for the formulas for LU-factorization that were originally given in the thesis of Lyon [], the derivation of an explicit algorithm for ULV factorization and related Moore-Penrose inversion, a complexity analysis and a short account of the connection between the HSS and the SSS (sequentially semi-separable) case. A direct consequence of the computational theory is that from a mathematical point of view the HSS structure is ‘closed’ for a number operations. The HSS complexity of a Moore-Penrose inverse equals the HSS complexity of the original, for a sum and a product of operators the HSS complexity is no more than the sum of the individual complexities.

We consider arbitrary families of unbounded normal operators, commuting in a strong sense, in particular algebras consisting of unbounded normal operators, and investigate their connections with some algebras of fractions of continuous functions on compact spaces. New examples and properties of spaces of fractions are also given.