On properties of Bernstein-type operator polynomials that approximate the Urysohn operator

We generalize properties of Bernstein polynomials to Bernstein-type operator polynomials that approximate the Urysohn operator.__________Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 9, pp. 1172–1181, September, 2004.     

On the spectral properties of an operator pencil

For a broad class of iterative algorithms for solving saddle point problems, the study of the convergence and of the optimal properties can be reduced to an analysis of the eigenvalues of operator pencils of a special form. A new approach to analyzing spectral properties of pencils of this kind is proposed that makes it possible to obtain sharp estimates for the convergence rate.     

Higher order differentiability properties of the composition and of the inversion operator

This paper contains theorems of r-th order Fréchet differentiability, with r≥1, for the autonomous composition operator and for the inversion operator in Schauder spaces. The optimality of the differentiability theorems for the composition is indicated by means of an ‘inverse result’. A main point of this paper is that (higher order) ‘sharp’ differentiability theorems for the composition operator can be proved by approximating the operator by composition operators whose superposing function is a polynomial, an idea which may be employed in other function space settings.     

On the lattice of all totally composition formations of finite groups

It is shown that the lattice of all totally composition formations of finite groups is algebraic.

     

Eigenfunctions of the hyperbolic composition operator

Letu inH ² be zero at one of the fixed points of a hyperbolic Möbius transform of the unit diskD. We will show, under some additional conditions onu, that the doubly cyclic subspaceS _u =V _n=– C ⁿ u contains nonconstant eigenfunctions of the composition operatorC . This implies that the cyclic subspace generated byu is not minimal. If there is an infinite dimensional minimal invariant subspace ofC (which is equivalent to the existance of an operator with only trivial invariant subspaces), then it is generated by a function with singularities at the fixed points of .     

On shape-preserving properties and simultaneous approximation of Stancu operator

In the present paper,the shape-preserving properties and the monotonicity for convex functions of Stancu operator are given.Moreover,the simultaneous approximation problems of this operator are also considered.     

On certain properties of the relative entropy of states of operator algebras

On leave of absence from the Mathematical Institute HAS, Budapest. This research was partially supported by the Hungarian National Foundation for Scientific Research grant No. 1171     

Weighted composition operator on the Fock space

We characterize the boundedness and compactness of a weighted composition operator on the Fock space. Our results use a certain integral transform. We also estimate the essential norm of a weighted compositon operator. The result could be extended to the higher-dimensional case.

     

On the Sturm-Liouville operator

We consider the homogeneous Sturm-Liouville differential equation, prove the existence of two linearly independent solutions that form a fundamental system, and obtain some of their properties.     

Differential properties of the Dirichlet-Neumann operator

Under consideration is the Dirichlet-Neumann operator on the boundary of the distorted half-space bounded below by a surface which is close to a hyperplane. We establish the differential properties of the operator depending on the smoothness of the boundary and find an expression for it.     

On the Libera operator

We show that the Libera operator, L, on some spaces of analytic functions is a continuous extension of the conjugate of the Cesàro operator. Results on L acting on various spaces are obtained. In particular, L maps the Bloch space into BMOA. We also prove some results on the best approximation by polynomials in Hardy and Bergman spaces.