Volume 59 , issue 3 ( 2007 ) | back |

Some subsets of ideal topological spaces | 75$-$84 |

**Abstract**

In ideal topological spaces, $\star$-dense in itself subsets are used to characterize ideals and mappings. In this note, properties of ${\cal A}_{\cal I}$-sets, ${\cal I}$-locally closed sets and almost strong ${\cal I}$-open sets are discussed. We characterize codense ideals by the collection of these sets. Also, we give a decomposition of continuous mappings and deduce some well-known results.

**Keywords:** Codense ideal, semiopen set,
preopen set, ${\cal I}$-locally closed set, $f_{\cal I}$-set, regular
${\cal I}$-closed set, ${\cal A}_{\cal I}$-set, semicontinuity, ${\cal A}_{\cal I}$-continuity,
$f_{\cal I}$-continuity.

**MSC:** 54A05, 54A10, 54C08, 54C10

Relations between some topologies | 85$-$95 |

**Abstract**

Generalizations of openness, such as semi-open, preopen, semi-pre-open, $\alpha$-open, etc\. are important in topological spaces and in particular in topological spaces on which ideals are defined. $\alpha$-equivalent topologies and $*$-equivalent topologies with respect to an ideal have some common properties. Relations between these aforementioned notions of openness are investigated within the framework of $\alpha$-equivalence and $*$-equivalence.

**Keywords:** $\alpha$-equivalent topologies; $*$-equivalent topologies; semi-open sets, preopen sets,
semi-pre-open sets, $\alpha$-open sets.

**MSC:** 54A05, 54A10, 54C08, 54C10

Invertible composition operators on Banach function spaces | 97$-$111 |

**Abstract**

In this paper, we relate composition operators with multiplication operators on the general Banach function spaces on a $\sigma$-finite measure space. We use this relation to study the invertibility and Fredholmness properties of composition operators on the Banach function spaces.

**Keywords:** Banach function spaces, closed range, composition operators,
Fredholm operator, invertibility, measurable transformation.

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

On a type of compactness via grills | 113$-$120 |

**Abstract**

In this paper, we introduce and study the idea of a new type of compactness, defined in terms of a grill $\cal G$ in a topological space $X$. Calling it $\cal G$-compactness, we investigate its relation with compactness, among other things. Analogues of Alexender's subbase theorem and Tychonoff product theorem are also obtained for $\cal G$-compactness. Finally, we exhibit a new method, in terms of the deliberations here, for construction of the well known one-point compactification of a $T_2$, locally compact and noncompact topological space.

**Keywords:** Grill, $\cal G$-compactness, $\cal G$-regularity, one-point compactification.

**MSC:** 54D35, 54D99

Signed degree sets in signed 3-partite graphs | 121$-$124 |

**Abstract**

If each edge of a 3-partite graph is assigned a positive or a negative sign then it is called a signed 3-partite graph. Also, signed degree of a vertex x in a signed 3-partite graph is the number of positive edges incident with x less than the number of negative edges incident with x. The set of distinct signed degrees of the vertices of a signed 3-partite graph is called its signed degree set. In this paper, we prove that every set of n integers is the signed degree set of some connected signed 3-partite graph.

**Keywords:** Signed graph, signed tripartite graph, signed degree, signed set.

**MSC:** 05C22

Weyl's and Browder's theorem for an elementary operator | 135$-$142 |

**Abstract**

Let $\cal H$ be a separable infinite dimensional complex Hilbert space and let $B(\cal H)$ denote the algebra of bounded operators on $\cal H$ into itself. The generalized derivation $\delta_{A,B}$ is defined by $\delta_{A,B}(X)=AX-XB$. For pairs $C=(A_{1},A_{2})$ and $D=(B_{1},B_{2})$ of operators, we define the elementary operator $\Phi_{C,D}$ by $\Phi_{C,D}(X)=A_{1}XB_{1}-A_{2}XB_{2}$. If $A_{2}=B_{2}=I$, we get the elementary operator $\Delta_{A_{1},B_{1}}(X)=A_{1}XB_{1}-X$. Let $d_{A,B}=\delta_{A,B}$ or $\Delta_{A,B}$. We prove that if $A, B^{*}$ are $\log$-hyponormal, then $f(d_{A,B})$ satisfies (generalized) Weyl's Theorem for each analytic function $f$ on a neighborhood of $\sigma(d_{A,B})$, we also prove that $f(\Phi_{C,D})$ satisfies Browder's Theorem for each analytic function $f$ on a neighborhood of $\sigma(\Phi_{C,D})$.

**Keywords:** Elementary Operators, $p$-hyponormal, $\log$-hyponormal, Weyl's Theorem, single
valued extension property.

**MSC:** 47B47, 47A30, 47B20

Some curvature conditions of the type $4\times2$ on the submanifolds satisfying Chen's equality | 143$-$150 |

**Abstract**

Submanifolds of the Euclidean spaces satisfying equality in the basic Chen's inequality have, as is known, many interesting properties. In this paper, we discuss the curvature conditions of the form $E\cdot S=0$ on such submanifolds, where $E$ is any of the standard 4-covariant curvature operators, $S$ is the Ricci curvature operator, and $E$ acts on $S$ as a derivation.

**Keywords:** Submanifolds, curvature conditions, basic equality.

**MSC:** 53B25, 53C40

Corrigendum to: Spline-wavelet solution of singularly perturbed boundary problem | 151 |

**Abstract**

Due to a technical error, Table 3 in the paper ``Spline-wavelet solution of singularly perturbed boundary problem'' by Desanka Radunović, published in the last issue of Matematčki Vesnik {\bf59}, 1--2 (2007), 31--46, appeared wrongly. The right Table 3 is printed here.

**Keywords:** Boundary layers, spline wavelets, collocation.

**MSC:** 65L10, 65L60, 65T60