S. Mercourakis, S., G. Vassiliadis.
An extension of Lorentz's almost convergence and applications in Banach spaces
(pp. 71-98)
A B S T R A C T S
DOMAINE NUMÉRIQUE DU PRODUIT AB AVEC A NORMAL
Mohamed Chraïbi Kaadoud
chraibik@ucam.ac.ma
2000 Mathematics Subject Classification: 18B30,
47A12.
Key words:
Domaine numérique, opérateur normal.
Let A, B be two linear operators on a complex
Hilbert space H. We extend a Bouldin's result (1969) conserning W(AB) -
the numerical range of the product AB. We show,
when AB = BA and A is normal, than
|
|
W(AB)
|
⊂ |
{< Ax,x > < Bx,x > || x||} = 1
|
|
|
PIECEWISE CONVEX CURVES AND THEIR INTEGRAL REPRESENTATION
M. D. Nedelcheva
mariana@mnet.bg
2000 Mathematics Subject Classification: 52A10.
Key words:
Convex arcs, Convex curves, Piecewise convex curves.
A convex arc in the plane is introduced as an oriented arc G
satisfying the following condition: For any three of its points
c1 < c2 < c3 the triangle c1c2c3 is counter-clockwise
oriented. It is proved that each such arc G is a closed and connected
subset of the boundary of the set FG
being the convex hull of G.
It is shown that the convex arcs are rectifyable and
admit a representation in the natural parameter
by the Riemann-Stieltjes integral with respect to
an increasing, nonnegative and continuous from the right
function s+. Further it is shown that the obtained representation
relates to the support function of the set FG.
Concerning the reverse question, namely what can be said for the curves
that admit such representation, it is shown that they
are exactly the curves that can be decomposed into
finitely many convex arcs. This result suggests the name
piecewise convex curves.
In particular, the class of piecewise convex curves
contains the convex curves being boundary sets of convex figures,
therefore the results from the paper can be used as a tool
for studying convex curves.
ON THE RANGE AND THE KERNEL OF DERIVATIONS
Said Bouali
bouali.said1@caramail.com,
Youssef Bouhafsi
ybouhafsi@yahoo.fr
2000 Mathematics Subject Classification:
Primary 47B47, 47B10; Secondary 47A30.
Key words:
Finite operator, n-multicyclic hyponormal operator.
Let H be a separable infinite
dimensional complex Hilbert space and let L(H) denote the algebra of all
bounded linear operators on H into itself.
Given A ∈ L(H),
the derivation δA : L(H)→ L(H) is defined
by δA(X) = AX-XA. In this paper we prove that if A is an
n-multicyclic hyponormal operator and T is hyponormal such that AT = TA,
then
||
δA(X)+T|| ≥ ||T|| for all X ∈ L(H). We establish the
same inequality if A is a finite operator and commutes with
normal operator T. Some related results are also given.
LITTLE G. T. FOR lp-LATTICE SUMMING OPERATORS
Lahcène Mezrag
lmezrag@caramail.com
2000 Mathematics Subject Classification:
46B28, 47D15.
Key words:
Banach lattice, completely bounded operator, convex operator,
lp-lattice summing operator, operator space.
In this paper we introduce and study the lp-lattice summing operators
in the category of operator spaces which are the analogous
of p-lattice
summing operators in the commutative case. We study
some interesting
characterizations of this type of operators which
generalize the results of
Nielsen and Szulga and we show that
Λ l∞( B(H) ,OH) ≠ Λ l2( B( H) ,OH) ,
in opposition to the commutative case.
CRITERION OF NORMALITY OF THE COMPLETELY REGULAR TOPOLOGY
OF SEPARATE CONTINUITY
Yakov S. Grinshpon
grinshpon@mail.ru
2000 Mathematics Subject Classification:
54C10, 54D15, 54G12.
Key words:
Separate continuity, normality, collectionwise normality,
scattered spaces, Cech-complete spaces, zero-dimensional spaces,
paracompactness, locally compact spaces.
For given completely regular topological spaces X and Y,
there is a completely regular space
such that for any completely regular space Z a mapping
f : X×Y® Z
is separately continuous if and only if
is continuous. We prove a necessary condition of normality, a
sufficient condition of collectionwise normality, and a criterion of
normality of the products
in the case when at least one factor is scattered.
ON LOCAL UNIFORM TOPOLOGICAL ALGEBRAS
Ali Oukhouya
aoukhouya@hotmail.com
2000 Mathematics Subject Classification:
Primary 46H05, 46H20; Secondary 46M20.
Key words:
Topological algebra, spectrum, Gel'fand map, Gel'fand
transform algebra, Gel'fand sheaf, local algebra, partitions of
unity.
Every unital ``combinatorially regular'' commutative
uniform complete locally m-convex algebra is local.
AN EXTENSION OF LORENTZ'S ALMOST CONVERGENCE AND APPLICATIONS
IN BANACH SPACES
S. Mercourakis
smercour@math.uoa.gr,
G. Vassiliadis
2000 Mathematics Subject Classification:
Primary 40C99, 46B99.
Key words:
Almost convergence, Banach limit, weakly
Cauchy sequence, independent sequence, uniform distribution of sequences.
We investigate an extension of the almost convergence of G. G. Lorentz
requiring that the means of a bounded sequence converge uniformly on a
subset M of N. We also present examples of sequences
α∈ l∞(N)
whose sequences of translates (Tn α)n≥ 0 (where T is the
left-shift operator on l∞(N)) satisfy:
(a) Tn α, n ≥ 0 generates a subspace E(α) of
l∞(N) that is
isomorphically embedded into c0 while α is not almost convergent.
(b) Tn α, n ≥ 0 admits an l1-subsequence and a nontrivial weakly
Cauchy subsequence while a is almost convergent.
Finally we show that, in the sense of measure, for almost all real sequences
taking values in a compact set K ⊆ R (with at least two
points), the sequence
(Tn α)n ≥ 0 is equivalent in the supremum norm to the usual
l1-basis and (hence) not almost convergent.
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