Multiplicative and additive D-stability, diagonal stability, Schur Dstability, H-stability are classical concepts which arise in studying linear dynamical systems. We unify these types of stability, as well as many others, in one concept of (D, G, •)-stability, which depends on a stability region D ⊂ C, a matrix class G and a binary matrix operation •. This approach allows us to unite several well-known matrix problems and to consider common methods of their analysis. In order to collect these methods, we make a historical review, concentrating on diagonal and D-stability. We prove some elementary properties of (D, G, •)-stable matrices, uniting the facts that are common for many partial cases. Basing on the properties of a stability region D which may be chosen to be a concrete subset of C (e.g. the unit disk) or to belong to a specified type of regions (e.g. LMI regions) we briefly describe the methods of further development of the theory of (D, G, •)-stability. We mention some applications of the theory of (D, G, •)-stability to the dynamical systems of different types.

The existence of the second (according to the module) eigenvalueλ2of a completely continuous nonnegative operatorAis proved under the conditions thatAacts in the spaceLp(Ω)orC(Ω)and its exterior squareA∧Ais also nonnegative. For the case when the operatorsAandA∧Aare indecomposable, the simplicity of the first and second eigenvalues is proved, and the interrelation between the indices of imprimitivity ofAandA∧Ais examined. For the case whenAandA∧Aare primitive, the difference (according to the module) ofλ1andλ2from each other and from other eigenvalues is proved.

Abstract. In this paper, eventually totally positive matrices (i.e. matrices all whose powers starting at some point are totally positive) are studied. We present a new approach to eventual total positivity which is based on the theory of eventually positive matrices. We mainly focus on the spectral properties of such matrices. We also study eventually J-sign-symmetric matrices and matrices, whose powers are P -matrices.Mathematics subject classification (2010): Primary 15A48; Secondary 15A18, 15A75.

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We prove that for any circulant matrix C of size n × n with the monic characteristic polynomial p(z), the spectrum of its (n − 1) × (n − 1) submatrix Cn−1 constructed with first n − 1 rows and columns of C consists of all critical points of p(z). Using this fact we provide a simple proof for the Schoenberg conjecture recently proved by R. Pereira and S. Malamud. We also prove full generalization of a higher order Schoenberg-type conjecture proposed by M. de Bruin and A. Sharma and recently proved by W.S.