On semigroups generated by differential operators on Lie groups

Let dR be the differential of a strongly continuous representation of a Lie group G on a Hilbert space H. Let P be the left-invariant second-order differential operator on G with positive semidefinite main part P₂ and with first-order part P₁. If Im(P₁) is in some sense subordinate to P₂ then dR(P) is a pregenerator of a strongly continuous semigroup of operators in H. If the whole P₁ is in some sense subordinate to P₂ then that semigroup is holomorphic or, even more, dR(P) is a pregenerator of a cosine operator function.     

Differential polynomials generated by linear differential equations

This article is devoted to considering value distribution theory of differential polynomials generated by solutions of linear differential equations in the complex plane. In particular, we consider normalized second-order differential equations f″+A(z)f=0, where A(z) is entire. Most of our results are treating the growth of such differential polynomials and the frequency of their fixed points, in the sense of iterated order.     

On the common part of the spectrum of finite difference operators generated by systems of polynomials

In this paper, we study the spectra of finite difference operators generated by systems of multiple orthogonal polynomials and the corresponding systems of measures of Stieltjes type. We show that the common support of the orthogonality measures coincides with the intersection of the spectra of the family of finite difference operators with common collection of Weyl functions.     

On the complex oscillation of differential polynomials generated by meromorphic solutions of differential equations in the unit disc

In this paper, we investigate the complex oscillation of differential polynomials generated by meromorphic solutions of differential equations

On quadratic fields generated by polynomials

Let be a polynomial of degree d ≥ 2 without multiple roots. Under the assumption of the ABC-conjecture, an asymptotic formula for the number of distinct fields among for has recently been given by Cutter, Granville, and Tucker. We use bounds for character sums to obtain an unconditional lower bound on the number of such fields for . Received: 19 November 2007     

Positive self-adjoint operators generated by products of differential expressions

Under the assumption of the product τ:=l⁺l of the regular differential expression l and its adjoint l⁺, being well-formed, a complete characterization of all the positive self-adjoint extensions of the minimal operator generated by τ in terms of boundary conditions is given.     

On algebras generated by positive operators

We study algebras generated by positive matrices, i.e., matrices with nonnegative entries. Some of our results hold in more general setting of vector lattices. We reprove and extend some theorems that have been recently shown by Kandi? and ?ivic. In particular, we give a more transparent proof of their result that the unital algebra generated by positive idempotent matrices E and F such that \(E F \ge F E\) is equal to the linear span of the set \(\{I, E, F, E F, F E, E F E, F E F, (E F)^2, (F E)^2\}\), and so its dimension is at most 9. We give examples of two positive idempotent matrices that generate unital algebra of dimension 2n if n is even, and of dimension \((2n - 1)\) if n is odd. We also prove that the algebra generated by positive matrices \(B_1\), \(B_2, \ldots , B_k\) is triangularizable if \(A B_i \ge B_i A\) (\(i=1,2, \ldots , k\)) for some positive matrix A with distinct eigenvalues.     

On topologies generated by some operators

The paper concerns topologies introduced in a topological space (X, τ) by operators which are much weaker than the lower density operators. Some properties of the family of sets having the Baire property and the family of meager sets with respect to such topologies are investigated.     

On semiiterative methods generated by Faber polynomials

Numerische Mathematik - Consider a nonsingular system of linear algebraic equationsA x=b, or in fixed point formx=Tx+c, where the eigenvalues ofT are contained in some compact subset Ω of the...     

On the norms of generalized translation operators generated by Dunkl type operators

An integral representation for the generalized translation operators generated by Dunkl type operators $$ \Lambda f(x) = f'(x) + \frac{{A'(x)}}{{A(x)}}\frac{{f(x) - f\left( { - x} \right)}}{2} $$ In the spaces $ {L_p}\left( \mathbb{R} \right) $ with weight A is established, and a known estimate for their norms is improved. It is proved that under some natural assumptions on the functions A, these norms do not exceed two. Bibliography: 24 titles.     

On stochastic processes generated by quadric operators

This paper is a continuation of our earlier paper (J. Theoret. Prob.3, 51–70). The existence and uniqueness of solutions of equations for quadric stochastic processes will be studied in this paper.     

On Markov operators preserving polynomials

The paper is concerned with a special class of positive linear operators acting on the space C(K)$C(K)$ of all continuous functions defined on a convex compact subset K   of R^d${\mathbf{R}}^d$, d?1$d?1$, having non-empty interior. Actually, this class consists of all positive linear operators T   on C(K)$C(K)$ which leave invariant the polynomials of degree at most 1 and which, in addition, map polynomials into polynomials of the same degree. Among other things, we discuss the existence of such operators in the special case where K is strictly convex by also characterizing them within the class of positive projections. In particular we show that such operators exist if and only if ∂K   is an ellipsoid. Furthermore, a characterization of balls of R^d${\mathbf{R}}^d$ in terms of a special class of them is furnished. Additional results and illustrative examples are presented as well.     

Generalized Humbert polynomials and second-order partial differential operators

We define a new class of Humbert's polynomials which generalizes the well-known class of Gegenbauer, Legendre, Pincherle, Horadam-Mahon, Kinney, Hordam-Pethe, Gould and Pathan-Khan polynomials. We prove that these generalized polynomials are eigenfunctions of the Hamiltonian of the Calogero-Sutherland model. We show several differential equations having these polynomials for solutions.     

On the solvable operators generated by uniformly correct problems

A first order differential-operator equation with an operator-coefficient generating a C₀ class semi-group is studied. Boundary conditions for which this equation possesses a unique solution dependent continuously on its right-hand side are derived. Two theorems are formulated and proved.     

On the spectrum of evolution operators generated by hyperbolic systems

We prove that the asymptotic behavior of semigroups generated by mixed-initial value problems for hyperbolic systems in one space variable is determined by the zeroes of the characteristic equation. The proof is based on the following reduction principle: we construct a reduced system (which is much simpler than the given one) such that the difference of the semigroups is a compact operator. The construction of the reduced system is presented in the non-autonomous case because it has implifications in the study of systems which are time periodic.