In the present article, we establish a definition of atomic systems in Krein spaces, specifically, fundamental tools theory formalism spaces and give complete characterization them. We also show that do not depend on decomposition space.

We discuss an identity in abstract scattering theory which can be interpreted as an integer-valued version of the Birman-Krein formula.

We study Robin-to-Robin maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains in R, n > 2, with generalized Robin boundary conditions.

We prove an isoperimetric inequality for uniformly log-concave measures and for the uniform measure on a uniformly convex body. These inequalities imply the log-Sobolev inequalities proved by Bobkov and Ledoux [12] and Bobkov and Zegarlinski [13]. We also recover a concentration inequality for uniformly convex bodies, similar to that proved by Gromov and Milman [22].

The theory of matrix valued orthogonal polynomials (MVOP) was introduced by M. G. Krein in 1949. Systematically studied in the last 15 years.