The saturation class for linear combinations of Bernstein operators

In this paper we study the saturation class for the linear combinations of Bernstein operators. The characterization of the saturation class involving the modulus of smoothness is proved under certain assumption. Received: 1 September 2007     

Existence and extensions of positive linear operators

In this paper we develop some unified methods, based on the technique of the auxiliary sublinear operator, for obtaining extensions of positive linear operators. In the first part, a version of the Mazur-Orlicz theorem for ordered vector spaces is presented and then this theorem is used in diverse applications: decomposition theorems for operators and functionals, minimax theory and extensions of positive linear operators. In the second part, we give a general sufficient condition (an implication between two inequalities) for the existence of a monotone sublinear operator and of a positive linear operator. Some particular cases in which this condition becomes necessary are also studied. Dedicated to Prof. Romulus Cristescu on his 80th birthday     

On linear operators preserving the set of positive polynomials

Following the classical approach of Pólya-Schur theory [13] we initiate in this paper the study of linear operators acting on and preserving either the set of positive univariate polynomials or similar sets of non-negative and elliptic polynomials. To Vladimir Igorevich Arnold with admiration     

Chebyshev polynomials for operators

The problem of uniqueness of the Chebyshev polynomials for bounded linear operators on normed linear spaces is investigated. Herrn Professor Dr. Dr. h.c. Heinz K?nig zu seinem achtzigsten Geburtstag gewidmet     

Point localization and spectral theory for symmetric random operators

In a Hilbert space context, we propose a rather general notion of “random operators” which allows for taking stochastic limits. After establishing a connection with measurable fields of closed operators, we may speak of a spectral theory for symmetric random operators. Received: 18 December 2008     

The inverse problem of geometric and golden means of positive definite matrices

In this paper we prove that the inverse mean problem of geometric and golden means of positive definite matrices

Bicommutants of Toeplitz operators

In this paper we discuss an unusual phenomenon in the context of Toeplitz operators in the Bergman space on the unit disc: If two Toeplitz operators commute with a quasihomogeneous Toeplitz operator, then they commute with each other. In the Bourbaki terminology, this result can be stated as follows: The commutant of a quasihomogeneous Toeplitz operator is equal to its bicommutant. Received: 11 March 2008     

Compact Hypergroup Characters and Finite Universal Korovkin Sets in L 1-Hypergroup Algebra

Let K be a compact hypergroup.We investigate Trig(K), the linear span of coordinate functions of the irreducible representations of K. Contrary to the group case, Trig(K) endowed with the usual multiplication does not bear an algebra structure, but it has a natural normed algebra structure when it inherits the convolution from , the algebra of all bounded Radon measures. We characterize the center of the algebras , L _p (K) and Trig(K) respectively, and consequently we obtain, for a certain class of hypergroups, the correspondence between the structure space of the center of L ₁(K) and the center of Trig(K). As an application we study the existence of a finite universal Korovkin set w.r.t. positive operators in the center of L ₁(K), in particular in L ₁(K), whenever K is commutative. The author was partially supported by the Romanian Academy under the grant No. 22/2007.     

Mapping Properties of Some Singular Operators in Besov Type Subspaces of C( − 1, 1)

We study mapping properties, boundedness and invertibility of some Cauchy singular integral operators in a scale of pairs of Besov type subspaces of C( − 1, 1). Our results include all those already available in the literature.     

Algebraic Properties of Toeplitz Operators with Radial Symbols on the Bergman Space of the Unit Ball

In this paper, we discuss some algebraic properties of Toeplitz operators with radial symbols on the Bergman space of the unit ball in . We first determine when the product of two Toeplitz operators with radial symbols is a Toeplitz operator. Next, we investigate the zero-product problem for several Toeplitz operators with radial symbols. Also, the corresponding commuting problem of Toeplitz operators whose symbols are of the form is studied, where and φ is a radial function. Ze-Hua Zhou: supported in part by the National Natural Science Foundation of China (Grand Nos.10671141, 10371091).     

On the spectrum of multivariable weighted composition operators

In this paper we study the point spectrum of the operator

Overiterated Linear Operators and Asymptotic Behaviour of Semigroups

The results provided hereby are related to the asymptotic behaviour of certain strongly continuous semigroups, which may be expressed in terms of iterates of positive linear operators, in the sense of Altomare’s theory. We present some applications to concrete cases involving continuous and discrete type operators, namely the Beta and the Stancu operators.      

Blow-up and critical exponents for parabolic equations with non-divergent operators: dual porous medium and thin film operators

Our first basic model is the fully nonlinear dual porous medium equation with source

Maximal Abelian von Neumann Algebras and Toeplitz Operators with Separately Radial Symbols

This paper mainly concerns abelian von Neumann algebras generated by Toeplitz operators on weighted Bergman spaces. Recently a family of abelian w^*-closed Toeplitz algebras has been obtained (see [5,6,7,8]). We show that this algebra is maximal abelian and is singly generated by a Toeplitz operator with a “common” symbol. A characterization for Toeplitz operators with radial symbols is obtained and generalized to the high dimensional case. We give several examples for abelian von Neumann algebras in the case of high dimensional weighted Bergman spaces, which are different from the one dimensional case.     

C 0,γ-Regularity for Vector-Valued Minimizers of Quasilinear Functionals

We discuss the interior C ^0,γ-everywhere regularity for minimizers of quasilinear functionals of the type

Optimization by the Homogenization Method for Nonlinear Elliptic Dirichlet Systems

We study the existence of solutions of control problems relative to a nonlinear elliptic system with Dirichlet boundary conditions. In this problem, the control variables are the coefficients of the equations and the open set where they are posed. It is known that this class of problems has no solution in general, but using homogenization results about elliptic systems we show the existence of solutions when the controls are searched in a bigger set. These results are related to the selection of optimal materials and shapes.     

Regular Couplings of Dissipative and Anti-Dissipative Unbounded Operators,Asymptotics of the Corresponding Non-Dissipative Processes and the Scattering Theory

In this paper a triangular model of a class of unbounded non-selfadjoint K ^r-operators A presented as a coupling of dissipative and anti-dissipative operators in a Hilbert space with real absolutely continuous spectra and with different domains of A and A ^* is considered. The asymptotic behaviour of the corresponding non-dissipative processes T_tf = e^itAf, generated from the semigroups T_t with generators iA, as t → ± ∞ are obtained. The strong wave operators, the scattering operator for the couple (A^*, A) and the similarity of A and the operator of multiplication by the independent variable are obtained explicitly. The considerations are based on the triangular models and characteristic functions of A. Kuzhel for unbounded operators and the limit values of the multiplicative integrals, describing the characteristic function of the considered model. Partially supported by Grant MM-1403/04 of MESC and by Scientific Research Grant 27/25.02.2005 of Shumen University.     

Convergence of the Iterated Aluthge Transform Sequence for Diagonalizable Matrices II: λ-Aluthge Transform

Let λ ∈ (0, 1) and let T be a r × r complex matrix with polar decomposition T = U|T|. Then the λ-Aluthge transform is defined by

Additive maps preserving the minimum and surjectivity moduli of operators

Let be a complex Hilbert space and let be the algebra of all bounded linear operators on . We characterize additive maps from onto itself preserving different spectral quantities such as the minimum modulus, the surjectivity modulus, and the maximum modulus of operators. Received: 15 July 2008