On 28 August at 17:00 Katriin Pirk will defend her doctoral thesis “Diametral diameter two properties, Daugavet-, and ∆-points in Banach spaces” for obtaining the degree of Doctor of Philosophy (Mathematics),
Senior Research Fellow Rainis Haller, PhD, University of Tartu
Research Fellow Johann Langemets, PhD, University of Tartu
Associate Professor Trond Arnold Abrahamsen, PhD, University of Agder (Norway)
Professor Anna Helena Kamińska, Dr habil, University of Memphis (USA)
Full Professor Miguel Martín Suarez, PhD, University of Granada (Spain)
The Daugavet property is an extreme and well researched geometric property of Banach spaces. The peculiarity of Banach spaces with such property is that every slice of the unit ball is 2, i.e. that space has the diameter 2 property. Examples of such spaces are c_0, l_∞, C[0,1] ja L_1 [0,1]. Reflexive spaces, e.g. Hilbert spaces, or separable dual spaces, e.g. l_1, however, have slices of arbitrarily small diameter. The research on the classical diameter 2 properties suggests different versions of these properties and these are thoroughly studied in this thesis. One of the starting points of the thesis is a study, that considered Banach spaces in which every element of norm 1 is a Δ-point, that is, in which every element in a slice of the unit ball has an almost diametral point in that slice. According to a slightly different geometric characterisation a Banach space has the Daugavet property if every element x of norm 1 is a Daugavet-point, that is, every element of the unit ball can be approximated in that space with convex combinations of almost diametral points of x. In the current thesis, the relations between new diametral diameter 2 properties are explained, the stability and inheritance results of these properties are investigated. In addition, examples are presented of well-known Banach spaces where Daugavet- and Δ-points are the same. A thorough study of the existence of Daugavet- and Δ-points in absolute sums affirms, among other things, that the concepts Daugavet-point and Δ-point are different. Let it be remarked, that the choice of the symbol Δ to denote an important notion in the thesis is at least a partial tribute to the new Delta Centre of University of Tartu.