Volume 50 , issue 3$-$4 ( 1998 ) | back |

50 years of ``Matematički Vesnik'' | 1$-$3 |

Continuity of the essential spectrum in the class of quasihyponormal operators | 71$-$74 |

**Abstract**

Let $H$ be a separable Hilbert space. We write $\sigma (A)$ for the spectrum of $A\in B(H)$, $\sigma_w(A)$ for the Weyl spectrum and $\sigma_b(A)$ for the Browder spectrum. Operator $A\in B(H)$ is quasihyponormal if $A^*(A^*A-AA^*)A\ge 0$, i.e.\ $\| A^*Ax\|\le \|A^2x\|$, for every $x\in H$.

**Keywords:** Weyl spectrum, Browder spectrum, quasihyponormal operator,
continuity of the spectrum.

**MSC:** 47A53

On the compactness in fuzzy topological spaces in Sostak's sense | 75$-$81 |

**Abstract**

In this paper, we establish new definitions of smooth closure and smooth interior of a fuzzy set [3] under the name of fuzzy closure and fuzzy interior of a fuzzy set which satisfy almost all properties of the corresponding definitions in fuzzy topological spaces in Chang's sense. As a consequence of these definitions we show that the topological concepts, especially various types of compactness, can be presented in a simple way, with no additional hypotheses needed which occured in [3] and with no counterpart in [2, 4, 6].

**Keywords:** Smooth closure, smooth interior, fuuzy topological space.

**MSC:** 54A40, 54D30

Boundary behavior of subharmonic functions on the unit disk | 83$-$87 |

**Abstract**

In this paper, we prove some boundary properties of subharmonic functions in the open unit disk of the complex plane.

**Keywords:** Subharmonic function.

**MSC:** 30E25, 30D40, 31A05

Julia points and strong analytic normality | 89$-$91 |

**Abstract**

In this paper we prove that, if a function $f$ holomorphic on the open unit disk belongs to the class $JB^*$, then $f$ has no Julia points.

**Keywords:** Julia point, strong analytic normality.

**MSC:** 30D45

On separable subalgebras of Azumaya algebras | 93$-$97 |

**Abstract**

Let $A$ be an Azumaya $C$-algebra. Then the set of all commutative separable subalgebras of $A$ and the set of separable subalgebras $B$ such that $V_A(B)=V_B(B)$ are in a one-to-one correspondence, where $V_A(B)$ is the commutator subring of $B$ in $A$, and the set of all separable subalgebras of $A$ is a disjoint union of the Azumaya algebras in $A$ over a commutative separable subalgebra of~$A$. The results are used to compute splitting rings for an Azumaya skew group ring.

**Keywords:** Azumaya algebras, Galois
extensions, splitting rings, skew group rings.

**MSC:** 16S30, 16W20

On nearly paracompact spaces and nearly full normality | 99$-$104 |

**Abstract**

This paper is a continuation of the study of nearly paracompact spaces, initiated by Singal and Arya in~[5]. After suitably defining the generalized versions of normality and full normality in our setting, we achieve, as our final objective, an analogue of the celebrated theorem of A. H. Stone on paracompactness viz\. ``a Hausdorff topological spaces is paracompact iff it is fully normal''. Incidentally, in course of the deliberation, we obtain a few extended forms of certain well known results on paracompactness.

**Keywords:** Nearly paracompact spaces,
nearly fully normal spaces, nearly normal spaces, almost regular spaces.

**MSC:** 54D99

An application for the Chebyshev polynomials | 105$-$110 |

**Abstract**

Two sequences of polynomials for which all zeros, regardless of degree $n$, can be given by the following ``simple formulae'' $$ \Gamma_{n,m}(\xi)=\cot\(\dfrac{(\xi+m)\pi}n\)\quad\text{and}\quad \Delta_{n,m}(\xi)=\tan\(\dfrac{(\xi+m)\pi}n\)\quad(0<\xi<1) $$ ($n=1,2,\dots$; $m=0,1,\dots,n-1$ and $m\ne(n-1)/2$ when $\xi=1/2$ and $n$ is odd in the case of $\Delta_{n,m}$) are obtained from the linear combination of the Chebyshev polynomials of the first and second kind.

**Keywords:** Chebyshev pomynolials, polynomials, zeros of polynomials.

**MSC:** 33C45, 33C90

Fuzzy strongly preopen sets and fuzzy strong precontinuity | 111$-$123 |

**Abstract**

A new class of generalized fuzzy open sets, called fuzzy strongly preopen sets is introduced. Fuzzy strong precontinuous, fuzzy strongly preopen and fuzzy strongly preclosed mappings between fuzzy topological spaces are defined. Their properties and the relationships between these mappings and other mappings introduced previously are investigated.

**Keywords:** Fuzzy topology, fuzzy semiopen set,
fuzzy preopen set, fuzzy strongly semiopen set, fuzzy strongly preopen set,
fuzyy strong precontinuous mapping, fuzzy strongly preopen mapping, fuzzy
strongly preclosed mapping.

**MSC:** 54A40

On CB-compact, countably CB-compact and CB-Lindelöf spaces | 125$-$133 |

**Abstract**

By a (countably) CB-compact space we call a topological space each cover (respectively each countable cover) of which by open sets with compact boundaries contains a finite subcover. By a CB-Lindelöf space we call a topological space each cover of which by open sets with compact boundaries contains a countable subcover. Basic properties of these spaces and relations of these spaces to some other classes of topological spaces are studied.

**Keywords:** CB-compact space, CB-Lindelöf space.

**MSC:** 54D30, 54D20

Spectral properties of the Cauchy operator and the operator of logarithmic potential type on $L^2$ space with radial weight | 135$-$139 |

**Abstract**

We consider the Cauchy operator $C$ and the operator of logarithmic potential type $L$ on $L^2(D,d\mu)$, defined by $$ Cf(z)=-\dfrac1\pi\int_D\dfrac{f(\xi)}{\xi-z}\,d\mu(\xi),\quad Lf(z)=- \dfrac1{2\pi}\int_D\log|z-\xi|\,f(\xi)\,d\mu(\xi), $$ where $D$ is the unit disc in $C$, $d\mu(\xi)=h(|\xi|)\,dA$, $h\in L^{\infty}(0,1)$ is a function, positive a.e.\ on $(0,1)$ and $dA$ the Lebesgue measure on~$D$. We describe all eigenvectors and eigenvalues of these operators in terms of some operators acting on $L^2(I,d\nu)$ with $I=[0,1]$, $d\nu(r)=rh(r)\,dr$.

**Keywords:** Cauchy integral operator,
operator of logarithmic potential type, space with radial weight.

**MSC:** 47G10, 45C05