Oscillatory matrices were introduced in the seminal work of Gantmacher and Krein. An n × matrix A is called oscillatory if all its minors are nonnegative there exists a positive integer k such that positive. The smallest for which this holds exponent . Krein showed always smaller than or equal to − 1 important nontrivial problem determine exact value exponent. Here we use successive elementary ...

Let J be an involutive Hermitian matrix with signature (t, n− t), 0 ≤ t ≤ n, that is, with t positive and n− t negative eigenvalues. The Krein space numerical range of a complex matrix A of size n is the collection of complex numbers of the form ξ ∗JAξ ξ∗Jξ , with ξ ∈ Cn and ξ∗Jξ = 0. In this note, a class of tridiagonal matrices with hyperbolical numerical range is investigated. A Matlab progr...

According to classical results by M. G. Krein and L. de Branges, for every positive measure μ on the real line R such that ∫ R dμ(t) 1+t2 <∞ there exists a Hamiltonian H such that μ is the spectral measure for the corresponding canonical Hamiltonian system JX ′ = zHX. In the case where μ is an even measure from Steklov class on R, we show that the Hamiltonian H normalized by detH = 1 belongs to...

A b s t r a c t . Let ~A,B be the Krein spectral shift function for a pair of operators A, B, with C = A B trace class. We establish the bound f F(I~A,B()~)I ) d,~ <_ f F ( 1 5 1 c l , o ( ) , ) l ) d A = ~ [F(j) F ( j 1 ) ] # j ( C ) , j= l where F is any non-negative convex function on [0, oo) with F(O) = 0 and #j (C) are the singular values of C. The choice F(t) = t p, p > 1, improves a rece...

In this paper, a Krein-Milman  type theorem in $T_0$ semitopological cone is proved,  in general. In fact, it is shown that in any locally convex $T_0$ semitopological cone, every convex compact saturated subset is the compact saturated convex hull of its extreme points, which improves the results of Larrecq.

We consider the eigenvalue-eigenvector problem where p 1 p m?1 = r. We prove an analogue of the classical Gantmacher{Krein Theorem for the eigenvalue-eigenvector structure of STP matrices in the case where p i 1 for each i, plus various extensions thereof. A matrix A is said to be strictly totally positive (STP) if all its minors are strictly positive. STP matrices were independently introduced...

The aim of this note is to provide the complete characterization of the numerical range of linear operators on the 2-dimensional Krein space C.

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 some key results from convex analysis in the setting of topological groups and monoids. These include separation theorems, Krein-Milman type theorems, and minimax theorems.