Serdica Mathematical Journal
Volume 29, Number 2, 2003
C O N T E N T S

Kalenda, O. F. K.
A characterization of weakly Lindelöf determined Banach spaces
(pp. 95108)

Borisova, G. S.
The operators A_{g} = gA+[`(g)]A^{*}
for a class of nondissipative operators A with a
limit of the corresponding correlation function
(pp. 109140)

Johanis, M.
Approximation of Lipschitz mappings
(pp. 141148)

Azevedo, S. S.
A basis for Zgraded identities of matrices over infinite fields
(pp. 149158)

Chbani, Z., H. Riahi.
Variational principles for monotone and maximal bifunctions
(pp. 159166)

Laghdir, M., N. Benkenza.
Duality in constrained DCoptimization via Toland's duality
approach
(pp. 167176)

Kostov Vl. P.
On a stratification defined by real roots of polynomials
(pp. 177186)

Molina, M., M. Mota, A. Ramos.
On the extinction probability for bisexual
branching processes in varying environments
(pp. 187194)
A B S T R A C T S
A CHARACTERIZATION OF WEAKLY LINDELÖF DETERMINED BANACH SPACES
Ondrej F. K. Kalenda
kalenda@karlin.mff.cuni.cz
2000 Mathematics Subject Classification:
46B26, 46B03, 46B04.
Key words:
Weakly Linde\"of determined Banach space, projectional
resolution of the identity, complemented subspace, Corson compact
space, Valdivia compact space.
We prove that a Banach space X is weakly Lindelöf determined
if (and only if) each nonseparable Banach space isomorphic to a
complemented subspace of X has a projectional resolution of the
identity. This answers a question posed by S. Mercourakis and
S. Negrepontis and yields a converse of AmirLindenstrauss'
theorem. We also prove that a Banach space of the form C(K)
where K is a continuous image of a Valdivia compactum is weakly
Lindelöf determined if (and only if) each nonseparable Banach space
isometric to a subspace of C(K) has a projectional resolution of
the identity.
THE OPERATORS A_{g} = gA+[`(g)]A^{*}
FOR A CLASS OF NONDISSIPATIVE OPERATORS A WITH A
LIMIT OF THE CORRESPONDING CORRELATION FUNCTION
Galina S. Borisova
g.borisova@fmi.shubg.net
2000 Mathematics Subject Classification:
Primary 47A48, Secondary 60G12.
Key words:
Operator colligation, nondissipative
curve, correlation function, wave operator, scattering operator.
In this work we present the operators A_{g} = gA+[`(g)]A^{*} with
Âg = 1/2 in the case of an
operator A from the class of nondissipative operators generating
nonselfadjoint curves, whose correlation functions have a limit as
t®±¥. The asympthotics of the stationary curves
e^{itAg}f as t®±¥ onto the absolutely
continuous subspace of A_{g} are obtained. These
asymptotics and the obtained asymptotics in [9] of the
nondissipative curves e^{itA}f allow to construct the scattering
theory for the couples (A_{g},A) and (A,A_{g}). We
consider the basic terms from the scattering theory  wave
operators, a scattering operator and the question of a similarity
of A and A_{g}. We obtain explicitly the wave operators,
the scattering operator and the similarity of A and
A_{g}.
APPROXIMATION OF LIPSCHITZ MAPPINGS
Michal Johanis
johanis@karlin.mff.cuni.cz
2000 Mathematics Subject Classification:
46B03.
Key words:
Approximation, uniform Gâteaux differentiability,
Lipschitz mappings.
We prove that any Lipschitz mapping from a separable Banach space into any
Banach space can be approximated by uniformly Gâteaux differentiable
Lipschitz mapping.
A BASIS FOR ZGRADED IDENTITIES OF MATRICES OVER INFINITE FIELDS
Sergio S. Azevedo
sergios@ime.unicamp.br
2000 Mathematics Subject Classification:
16R10, 16R20, 16R50.
Key words:
Matrix algebra, variety of algebras,
polynomial identities, graded identities.
The algebra M_{n}(K) of the matrices n×n over a field K can
be regarded as a Zgraded algebra. In this paper, it is
proved that if K is an infinite field, all the Zgraded
polynomial identities of M_{n}(K) follow from the identities:



 

 

zyx,
a(x) = a(y) =
a(z), 

 

where a is the degree of the corresponding variable.
This is a generalization of a result of Vasilovsky about the Zgraded
identities of the algebra M_{n}(K) over fields of characteristic 0.
VARIATIONAL PRINCIPLES FOR MONOTONE AND MAXIMAL BIFUNCTIONS
Zaki Chbani
chbaniz@ucam.ac.ma,
Hassan Riahi
hriahi@ucam.ac.ma
2000 Mathematics Subject Classification:
49J40, 49J35, 58E30, 47H05.
Key words:
Brø ndstedRockafellar principle, equilibrium problem.
We establish variational principles for monotone and maximal bifunctions of
Brø ndstedRockafellar type by using our characterization of bifunction's
maximality in reflexive Banach spaces. As applications, we give an existence
result of saddle point for convexconcave function and solve an approximate
inclusion governed by a maximal monotone operator.
DUALITY IN CONSTRAINED DCOPTIMIZATION VIA TOLAND'S DUALITY
APPROACH
M. Laghdir
laghdir@ucd.ac.ma,
N. Benkenza
2000 Mathematics Subject Classification:
90C48, 49N15, 90C25.
Key words:
Conjugate, difference convex, TolandSinger
duality, DCconstraint.
In this paper we reconsider a nonconvex duality
theory established by B. Lemaire and M. Volle (see [4]),
related to a primal problem of minimizing the difference of two
convex functions subject to a DCconstraint. The purpose of this
note is to present a new method based on TolandSinger duality
principle. Applications to the case when the constraints are
vectorvalued are provided.
ON A STRATIFICATION DEFINED BY REAL ROOTS OF POLYNOMIALS
Vladimir Petrov Kostov
kostov@math.unice.fr
2000 Mathematics Subject Classification:
12D10.
Key words:
Multiplicity vector; multiplicity surplus.
We consider the family of polynomials
P(x,a) = x^{n}+a_{1}x^{n1}+¼+
a_{n},x,a_{i} Î R,
and the stratification of
R^{n} @
{ (a_{1},¼,a_{n})
a_{i} Î R}
defined by the
multiplicity vector of the real roots of P. We prove smoothness
of the strata and a transversality property of their
tangent spaces.
ON THE EXTINCTION PROBABILITY FOR BISEXUAL BRANCHING PROCESSES IN
VARYING ENVIRONMENTS
Manuel Molina
mmolina@unex.es,
Manuel Mota
mota@unex.es,
Alfonso Ramos
aramos@unex.es
2000 Mathematics Subject Classification:
60J80.
Key words:
Bisexual branching processes. Branching processes
in varying environments. Extinction probability.
In this paper, the bisexual branching process in varying
environments introduced in [9] is considered and some sufficient
conditions for the existence of positive probability of
nonextinction are established.
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