Banach space theory
Research focus
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The research group works on various aspects of the theory of Banach spaces connected with the
structure of the space, the geometry of the unit ball and measures of non-compactness, and the study
of operators on Banach spaces.
1. A first subject, mainly researched by P.Terenzi, is the basis problem. One of the
fundamental problems in Banach space theory was the question about the existence of a
Schauder basis in every separable Banach space. After P.Enflo solved the problem in the
negative, the interest moved on what kind of “basislike” systems one can find in an arbitrary
separable Banach spaces. The existence of uniformly minimal Markushevic-bases, i.e.
biorthogonal systems {xn , xn*}, {xn
*} total, Sup || xn|| ||xn
*|| < +¥, such that cl(span {xn}) =
X , and several properties of M-bases have been guaranteed in many previous papers
(R.I.Ovsepian, A.Pelcinsky, P.Terenzi) . Every x in X is uniquely represented by
? ¥
=1
*( )
n
n n x x x but, lacking a Schauder basis, to ensure norm convergence of the basis
expansion, one has to allow blockings and permutations. The aim of the research is to
control blockings and permutations to provide sequences {xn} whose properties make them
as close as possible to Schauder bases.
2. A second research line focuses on some topological properties in normed spaces, in
connection with fixed point theory and with the study of evenly convex sets, i.e sets which
are intersection of (any number of) open halfspaces.
The fixed point property(fpp) for continuous mappings characterizes compact sets among
convex subsets of a Banach space. Actually, analogous characterizations hold with smaller
classes of functions. For instance the fpp for Lipschitzian mappings does characterize
compact sets (Lin-Sternfeld, 1981) , while the fpp for non-expansive mappings (1-
Lipschitzian ) does not. If one considers continuous maps which control some measure of
non-compactness of sets, instead of controlling distances, Lipschitzian mappings are
replaced by m -contractive maps. For 1-m -contractive maps, [2] provides a positive answer.
Even convexity has been widely used in the last fifteen years in quasi-convex programming,
where it plays a relevant role in duality. In [V.Klee, E. Maluta, C. Zanco, "Basic properties
of evenly convex sets", Journal of Convex Analysis 14 (2007), No. 1, 137—148], a detailed
study of such sets is presented as well as results on characterization of sets which admit
only e-convex sections or projections, and the results are totally satisfactory only in low
dimensions. Nevertheless, the natural environment for e-convex sets are infinite dimensional
normed spaces, where the theory is still at the beginning, and the only results are in the
separable case.
3. As a third subject, E. Laeng has been working on three separate projects on sharp norm
inequalities (best constants) and dimension-free norm estimates. (1) He studies Fourier
multiplier operators related to the Hilbert Transform (in particular multipliers periodized in
the frequency domain and various versions of discrete Hilbert transform) trying to determine
the exact norm of these operators on LP, 1 Full ProfessorsDipartimento di afferenza
Docenti afferenti
Paolo Terenzi
Associate Professors
Enrico Laeng
Elisabetta Maluta