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...

As a corollary of the main resu!t of this paper (Theorem i) we show that if ~ i~ not S-dense, then I~[ ~< 1~ s [. This contains as special cases the results of Sauer [4] and Perles and Shelab [5], and of Karpovski and Milman [3]. Before stating Theorem l, we need one more definition; ~ c 3~ is monotone if f ¢ ~ , g e 3 ~ and f ~ g imply g ¢ ~ . Note ~.hat ~ s is monotone and that a monotone fam...

Sign type spectra are an important tool in the investigation of spectral properties of selfadjoint operators in Krein spaces. It is our aim to show that also sign type spectra for normal operators in Krein spaces provide insight in the spectral nature of the operator: If the real part and the imaginary part of a normal operator in a Krein space have real spectra only and if the growth of the re...

This paper extends Krein-Rutman's theorem on linear operators leaving an invariant cone in infinite-dimensional Banach spaces to multivalued convex functions. The result is applied to obtain necessary and sufficient conditions for global controllability and reachability of nonlinear discrete-time systems described by convex processes.

The Bernstein approximation problem is to determine whether or not the space of all polynomials is dense in a given weighted C0-space on the real line. A theorem of de Branges characterizes non-density by existence of an entire function of Krein class being related with the weight in a certain way. An analogous result holds true for weighted sup-norm approximation by entire functions of exponen...

We extend recent results by Pisier on K-subcouples, i.e. subcouples of an interpolation couple that preserve the K-functional (up to constants) and corresponding notions for quotient couples. Examples include interpolation (in the pointwise sense) and a reinter-pretation of the Adamyan-Arov-Krein theorem for Hankel operators.

The main purpose of this paper is to investigate the formal deficiency indices N±(I) of a symmetric first order system Jf ′ +Bf = λH f on an interval I, where I = R or I = R±. Here J,B,H are n × n matrix valued functions and the Hamiltonian H ≥ 0 may be singular even everywhere. We obtain two results for such a system to have minimal numbers N±(R) = 0 (resp. N±(R±) = n) and a criterion for thei...