Volume 57 , issue 3$-$4 ( 2005 ) | back |

Asymptotic planarity of Dresher mean values | 61$-$63 |

**Abstract**

A family of Dresher mean values is asymptotically planar with respect to its two parameters. An asymptotic formula presenting this property holds if: (a) all variables converge to the same value; and, equivalently, because of means homogeneity, (b) for variables with same additive increment converging to infinity.

**Keywords:** Dresher mean values, asymptotic behavior.

**MSC:** 26E60

The measure of noncompactness of matrix transformations on the spaces $c^p(\Lambda)$ and $c^p_{\infty}(\Lambda)$ ($1 | 65$-$78 |

**Abstract**

We study linear operators between certain sequence spaces X and Y when X is $C^{p}(\Lambda)$ or $C^{p}_{\infty}(\Lambda)$ and Y is one of the spaces: $c$, $c_{0}$, $l_{\infty}$, $c(\mu)$, $c_{0}(\mu)$, $c_{\infty}(\mu)$, that is, we give necessary and sufficient conditions for A to map X into Y and after that necessary and sufficient conditions for A to be a compact operator. These last conditions are obtained by means of the Hausdorff measure of noncompactness and given in the form of conditions for the entries of an infinite matrix A.

**Keywords:** Matrix transformations, compact operators, measure of noncompactness.

**MSC:** 40H05, 46A45

Some remarks about bounded derivations on the Hilbert space of square summable matrices | 79$-$85 |

**Abstract**

It is known that not every Banach algebra has non-trivial bounded derivations. For instance, consider large families of weighted semisimple Banach algebras. In particular, we will be concerned with derivations within the concrete frame of the non-abelian, non-unitary, involutive Banach algebra $l^{2}(N^{2})$. The theoretical interest in this algebra is based on the well-known fact that it is isomorphic to the class of Hilbert-Schmidt operators acting between two given separable Hilbert spaces. In this article, we characterize and determine the explicit structure of all bounded derivations on $l^{2}(N^{2})$.

**Keywords:** Bounded and unbounded derivations
on Banach, $C^{\ast}$ or von Neumann algebras; inner and outer derivations;
Hilbert-Schmidt operators.

**MSC:** 46H05, 46J45, 47B47

The Voronovskaya Theorem for generalized Baskakov-Kantorovich operators in polynomial weight spaces | 87$-$94 |

**Abstract**

Recently, K. Bogalska has applied a new technique to establish a Voronovskaya-type theorem. In this paper we will also prove the Voronovskaya Theorem for geenralized Baskakov-Kantorovich-type operators in polynomial weight spaces using the same approach.

**Keywords:** Voronovskaya Theorem, polynomial weighted spaces.

**MSC:** 41A36

Spectral multiplicity of certain Gaussian processes | 95$-$98 |

**Abstract**

In this paper we compare our conditions under which the spectral multiplicity of a Gaussian process given by an integral expression equals the one with the Cramer's regularity conditions.

**Keywords:** Gaussian stochastic process, canonical representation, spectral multiplicity,
equivalence of Gaussian measures.

**MSC:** 60G12

Limit theorem for high level $A$-upcrossings by $\chi$-field | 99$-$108 |

**Abstract**

A Poisson limit theorem for $A$-points of upcrossings of a high level by trajectories of the random field $\chi(t)$ is established. Kallenberg theorem, standard results from asymptotic methods of Gaussian process and fields and Piterbarg result of high level intersection of $\chi$-field are exploited.

**Keywords:** Gaussian vector field, Poisson field, $A$-points.

**MSC:** 60G15, 60G60

Compact composition operators on Lorentz spaces | 109$-$112 |

**Abstract**

We give a necessary and sufficient condition for the compactness of composition operators on the Lorentz spaces.

**Keywords:** Compact operator, composition operators, Lorentz spaces, measurable transformation.

**MSC:** 47B33, 46E30, 47B07, 46B70

On pseudo-sequence coverings, $\pi$-images of metric spaces | 113$-$120 |

**Abstract**

In this paper, we prove that a space $X$ is a pseudo-sequence-covering, $\pi$-image of a metric space if and only if $X$ has a point-star network consisting of $wcs$-covers, which answers a conjecture posed by Lin affirmatively. As an application of this result, we have that a space is a pseudo-sequence-covering, $\pi$-image of a separable metric space is characterized as a sequentially-quotient, $\pi$-image of a separable metric space.

**Keywords:** Metric space, $\pi$-mapping, pseudo-sequence-covering mapping, $wcs$-cover,
$cs^*$-cover, point-star network.

**MSC:** 54E35, 54E40

On strongly pre-open sets and a decomposition of continuity | 121$-$125 |

**Abstract**

In this paper some new concepts of generalized open sets and generalized continuous functions are studied. The notions of strongly preopen sets, $S$-precontinuous functions, $\delta$-continuous functions are introduced and some of their properties are studied showing their behavior in comparison to other generalized structures already available in literature. The final result in this paper gives one new decomposition od continuity.

**Keywords:** Strongly pre-open sets, $S$-precontinuous functions, $\delta$-continuous functions.

**MSC:** 54C10, 54A05