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. 95-108)
• 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. 109-140)
• Johanis, M. Approximation of Lipschitz mappings (pp. 141-148)
• Azevedo, S. S. A basis for Z-graded identities of matrices over infinite fields (pp. 149-158)
• Chbani, Z., H. Riahi. Variational principles for monotone and maximal bifunctions (pp. 159-166)
• Laghdir, M., N. Benkenza. Duality in constrained DC-optimization via Toland's duality approach (pp. 167-176)
• Kostov Vl. P. On a stratification defined by real roots of polynomials (pp. 177-186)
• Molina, M., M. Mota, A. Ramos. On the extinction probability for bisexual branching processes in varying environments (pp. 187-194)

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 non-separable 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 Amir-Lindenstrauss' 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 non-separable 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.shu-bg.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^itA_gf 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^itAf 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 Z-GRADED 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 Z-graded algebra. In this paper, it is proved that if K is an infinite field, all the Z-graded polynomial identities of M_n(K) follow from the identities:

x = 0,    |a(x)| ³ n, xy = yx,    a(x) = a(y) = 0, xyz = 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 Z-graded 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 h-riahi@ucam.ac.ma

2000 Mathematics Subject Classification: 49J40, 49J35, 58E30, 47H05. Key words: Brø ndsted-Rockafellar principle, equilibrium problem.

We establish variational principles for monotone and maximal bifunctions of Brø ndsted-Rockafellar type by using our characterization of bifunction's maximality in reflexive Banach spaces. As applications, we give an existence result of saddle point for convex-concave function and solve an approximate inclusion governed by a maximal monotone operator.

DUALITY IN CONSTRAINED DC-OPTIMIZATION 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, Toland-Singer duality, DC-constraint.

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 DC-constraint. The purpose of this note is to present a new method based on Toland-Singer duality principle. Applications to the case when the constraints are vector-valued 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) = xⁿ+a₁x^n-1+¼+ a_n,x,a_i Î R, and the stratification of Rⁿ @ { (a₁,¼,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 non-extinction are established.