Volume 1, Issue 2013 -

An overview of some recent developments on the Invariant Subspace Problem This paper presents an account of some recent approaches to the Invariant Subspace Problem. It contains a brief historical account of the problem, and some more detailed discussions of specific topics, namely, universal operators, the Bishop operators, and Read’s Banach space counter-example involving a finitely strictly singular operator.

On the Normality of the Unbounded Product of Two Normal Operators Let A and B be two -non necessarily bounded- normal operators. We give new conditions making their product normal. We also generalize a result by Deutsch et al on normal products of matrices.

In this paper we consider the truncated shift operator S u on the model space K 2 u := H 2 θ uH 2 . We say that a complex number λ is an extended eigenvalue of Su if there exists a nonzero operator X, called extended eigenvector associated to λ, and satisfying the equation S u X = λXS u . We give a complete description of the set of extended eigenvectors of S u , in the case of u is a Blaschke product..

We study the homogeneous Riemann-Hilbert problem with a vanishing scalar-valued continuous coefficient. We characterize non-existence of nontrivial solutions in the case where the coefficient has its values along several rays starting from the origin. As a consequence, some results on injectivity and existence of eigenvalues of Toeplitz operators in Hardy spaces are obtained.

In the beginning of the twentieth century, Plemelj introduced the notion of factorization of matrix functions. The matrix factorization finds applications in many fields such as in the diffraction theory, in the theory of differential equations and in the theory of singular integral operators. However, the explicit formulas for the factors of the factorization are known only in a few classes of matrices. In the present paper we consider a new approach to obtain the factorization of a rational matrix function, relative to the unit circle. The constructed method is based on the relation between the general solution of a homogeneous Riemann-Hilbert problem and a solution of a linear system of difference equations with constant coefficients.