Trond Arnold Abrahamsen

• Abrahamsen, Trond A.; Hájek, Petr; Lima, Vegard & Troyanski, Stanimir (2026). Strictly convex norms and the local diameter two property. Revista Matemática Complutense. ISSN 1139-1138. 39, p. 857–877. doi: 10.1007/s13163-025-00559-z. Full text in Research Archive Show summary

  We introduce and study a strict monotonicity property of the norm in solid Banach lattices of real functions that prevents such spaces from having the local diameter two property. Then we show that any strictly convex 1-symmetric norm on $$c_0(\Gamma )$$ c 0 ( Γ ) possesses this property. In the opposite direction, we show that any Banach space which is strictly convex renormable and contains a complemented copy of $$c_0({\mathbb {N}}),$$ c 0 ( N ) , admits an equivalent strictly convex norm for which the space has the local diameter two property. In particular, this enables us to construct a strictly convex norm on $$c_0(\Gamma ),$$ c 0 ( Γ ) , where $$\Gamma $$ Γ is uncountable, for which the space has a 1-unconditional basis and the local diameter two property.

• Abrahamsen, Trond Arnold; Aliaga, Ramon J.; Lima, Vegard; Martiny, Andre; Perreau, Yoël & Prochazka, Antonín [Show all 7 contributors for this article] (2024). A relative version of Daugavet points and the Daugavet property. Studia Mathematica. ISSN 0039-3223. 279(3), p. 191–241. doi: 10.4064/sm240118-2-8. Full text in Research Archive
• Abrahamsen, Trond Arnold; Aliaga, Ramon J.; Lima, Vegard; Martiny, Andre; Perreau, Yoël & Prochazka, Antonín [Show all 7 contributors for this article] (2024). Delta-points and their implications for the geometry of Banach spaces. Journal of the London Mathematical Society. ISSN 0024-6107. 109(5). doi: 10.1112/jlms.12913. Full text in Research Archive
• Abrahamsen, Trond Arnold; Lima, Vegard; Martiny, Andre & Perreau, Yoël (2022). Asymptotic geometry and Delta-points. Banach Journal of Mathematical Analysis. ISSN 2662-2033. 16(4). doi: 10.1007/s43037-022-00210-9. Full text in Research Archive Show summary

  We study Daugavet- and Δ-points in Banach spaces. A norm one element x is a Daugavet-point (respectively, a Δ-point) if in every slice of the unit ball (respectively, in every slice of the unit ball containing x) you can find another element of distance as close to 2 from x as desired. In this paper, we look for criteria and properties ensuring that a norm one element is not a Daugavet- or Δ-point. We show that asymptotically uniformly smooth spaces and reflexive asymptotically uniformly convex spaces do not contain Δ-points. We also show that the same conclusion holds true for the James tree space as well as for its predual. Finally, we prove that there exists a superreflexive Banach space with a Daugavet- or Δ-point provided there exists such a space satisfying a weaker condition.

• Abrahamsen, Trond Arnold; Lima, Vegard & Martiny, Andre (2022). Delta-points in Banach spaces generated by adequate families. Illinois Journal of Mathematics. ISSN 0019-2082. 66(3), p. 421–434. doi: 10.1215/00192082-10123638. Full text in Research Archive
• Abrahamsen, Trond Arnold; Lima, Vegard; Martiny, Andre & Troyanski, Stanimir (2021). Daugavet- and delta-points in Banach spaces with unconditional bases. Transactions of the American Mathematical Society - Series B. 8(13), p. 379–398. doi: 10.1090/btran/68. Full text in Research Archive
• Abrahamsen, Trond Arnold; Haller, Rainis; Lima, Vegard & Pirk, Katriin (2020). Delta- and Daugavet points in Banach spaces. Proceedings of the Edinburgh Mathematical Society. ISSN 0013-0915. 63(2), p. 475–496. doi: 10.1017/S0013091519000567. Full text in Research Archive
• Abrahamsen, Trond Arnold; Guerrero, Julio Becerra; Haller, Rainis; Lima, Vegard & Poldvere, Mert (2020). Banach spaces where convex combinations of relatively weakly open subsets of the unit ball are relatively weakly open. Studia Mathematica. ISSN 0039-3223. 250(3), p. 297–320. doi: 10.4064/sm181016-10-1. Full text in Research Archive
• Abrahamsen, Trond Arnold; Hájek, Petr & Troyanski, Stanimir (2020). Almost square dual Banach spaces. Journal of Mathematical Analysis and Applications. ISSN 0022-247X. 487(2). doi: 10.1016/j.jmaa.2020.124003. Full text in Research Archive
• Abrahamsen, Trond Arnold; Dovland, Olav & Poldvere, Mert (2019). New applications of extremely regular function spaces. Pacific Journal of Mathematics. ISSN 0030-8730. 301(2). doi: 10.2140/pjm.2019.301.385. Full text in Research Archive Show summary

  Let L be an infinite locally compact Hausdorff topological space. We show that extremely regular subspaces of C0(L) have very strong diameter 2 properties and, for every real number ε with 0 <ε<1, contain an ε-isometric copy of c0. If L does not contain isolated points they even have the Daugavet property, and thus contain an asymptotically isometric copy of ℓ1.

• Abrahamsen, Trond Arnold; Dovland, Olav Kristian & Põldvere, Märt (2019). New applications of extremely regular function spaces. Pacific Journal of Mathematics. ISSN 0030-8730. 301(2), p. 385–394. doi: 10.2140/pjm.2019.301.385. Full text in Research Archive
• Abrahamsen, Trond Arnold; Fonf, Vladimir P.; Smith, Richard J. & Troyanski, Stanimir (2019). Polyhedrality and decomposition. Quarterly Journal of Mathematics. ISSN 0033-5606. 70(2), p. 409–427. doi: 10.1093/qmath/hay050. Full text in Research Archive
• Abrahamsen, Trond Arnold & Lima, Vegard (2018). Relatively weakly open convex combinations of slices. Proceedings of the American Mathematical Society. ISSN 0002-9939. 146(10), p. 4421–4427. doi: 10.1090/proc/14120. Full text in Research Archive
• Abrahamsen, Trond Arnold; Hájek, Petr; Nygaard, Olav Kristian & Troyanski, Stanimir (2018). Strongly Extreme Points and Approximation Properties. Canadian Mathematical Bulletin (CMB). ISSN 0008-4395. 61(3), p. 449–457. doi: 10.4153/CMB-2017-067-3. Full text in Research Archive
• Abrahamsen, Trond Arnold; Martiny, Andre; Leraand, Aleksander & Nygaard, Olav Kristian (2017). Two properties of Müntz spaces. Demonstratio Matematicae. ISSN 0420-1213. 50(1), p. 239–244. doi: 10.1515/dema-2017-0025. Full text in Research Archive Show summary

  We show that Müntz spaces, as subspaces of C[0, 1], contain asymptotically isometric copies of c0 and that their dual spaces are octahedral.

• Abrahamsen, Trond Arnold; Langemets, Johann & Lima, Vegard (2016). Almost square Banach spaces. Journal of Mathematical Analysis and Applications. ISSN 0022-247X. 434(2), p. 1549–1565. doi: 10.1016/j.jmaa.2015.09.060. Full text in Research Archive
• Lima, Vegard; Abrahamsen, Trond Arnold; Nygaard, Olav Kristian & Troyanski, Stanimir (2016). Diameter two properties, convexity and smoothness. Milan Journal of Mathematics. ISSN 1424-9286. 84(2), p. 231–242. doi: 10.1007/s00032-016-0258-1. Full text in Research Archive
• Abrahamsen, Trond Arnold; Hájek, Petr; Nygaard, Olav Kristian; Talponen, Jarno & Troyanski, Stanimir (2016). Diameter 2 properties and convexity. Studia Mathematica. ISSN 0039-3223. 232(3), p. 227–242. doi: 10.4064/sm8317-4-2016. Full text in Research Archive
• Abrahamsen, Trond Arnold (2015). Linear extensions, almost isometries, and diameter two. Extracta Mathematicae. ISSN 0213-8743. 30(2), p. 135–151. Full text in Research Archive
• Abrahamsen, Trond Arnold; Langemets, Johann; Lima, Vegard & Nygaard, Olav Kristian (2015). On thickness and thinness of Banach spaces. Houston Journal of Mathematics. ISSN 0362-1588. 41(1), p. 97–111. Full text in Research Archive
• Abrahamsen, Trond Arnold; Lima, Vegard & Nygaard, Olav Kristian (2014). Almost isometric ideals in Banach spaces. Glasgow Mathematical Journal. ISSN 0017-0895. 56(2), p. 395–407. doi: 10.1017/S0017089513000335. Full text in Research Archive
• Abrahamsen, Trond Arnold (2014). Strict u-ideals and u-summands in banach spaces. Mathematica Scandinavica. ISSN 0025-5521. 114(2), p. 216–225. doi: 10.7146/math.scand.a-17108. Full text in Research Archive
• Abrahamsen, Trond Arnold; Lima, Vegard & Dovland, Olav (2013). Remarks on diameter 2 properties. Journal of Convex Analysis. ISSN 0944-6532. Full text in Research Archive
• Abrahamsen, Trond Arnold; Lima, Vegard & Nygaard, Olav Kristian (2013). Remarks on Diameter 2 properties. Journal of Convex Analysis. ISSN 0944-6532. 20(2), p. 439–452. Full text in Research Archive Show summary

  If X is an infinite-dimensional uniform algebra, if X has the Daugavet property or if X is a proper M-embedded space, every relatively weakly open subset of the unit ball of the Banach space X is known to have diameter 2, i.e., X has the diameter 2 property. We prove that in these three cases even every finite convex combination of relatively weakly open subsets of the unit ball have diameter 2. Further, we identify new examples of spaces with the diameter 2 property outside the formerly known cases; in particular we observe that forming lp-sums of diameter 2 spaces does not ruin diameter 2 structure.

• Abrahamsen, Trond Arnold & Nygaard, Olav (2011). On λ-strict ideals in banach spaces. Bulletin of the Australian Mathematical Society. ISSN 0004-9727. 83(2), p. 231–240. doi: 10.1017/S0004972710001735. Full text in Research Archive
• Abrahamsen, Trond Arnold & Nygaard, Olav (2011). ON lambda-STRICT IDEALS IN BANACH SPACES. Bulletin of the Australian Mathematical Society. ISSN 0004-9727. 83(2), p. 231–240. doi: 10.1017/S0004972710001735. Full text in Research Archive
• Abrahamsen, Trond Arnold; Lima, Vegard & Lima, Åsvald (2009). UNCONDITIONAL IDEALS OF FINITE RANK OPERATORS II. Houston Journal of Mathematics. ISSN 0362-1588. 35(2), p. 627–646. Full text in Research Archive Show summary

  Let E be a subspace of a normed space F. It is known that E is an ideal (resp. a u-ideal) in F if and only if E is an ideal (resp. a u-ideal) in G for every subspace E subset of G subset of F in which E has finite codimension (resp. codimension <= 2). We show that in many cases a space of finite rank operators is an ideal (resp. a u-ideal) in a larger space if and only if it is an ideal (resp. a u-ideal) in a space in which it has codimension 1. In particular, we show that F(Y, X) is an ideal (resp. a u-ideal) in W (Y, X **) for all Banach spaces Y if and only if for every reflexive Banach space Y and T is an element of W (Y, X **), F(Y, X) is an ideal (resp. a u-ideal) in span (F(Y, X), {T}).

• Lima, Åsvald; Abrahamsen, Trond Arnold & Lima, Vegard (2008). Unconditional ideals of finite rank operators. Czechoslovak Mathematical Journal. ISSN 0011-4642. 58(4), p. 1257–1278. Full text in Research Archive
• Abrahamsen, Trond Arnold (2006). Weak metric approximation properties and nice projections. Acta et Commentationes Universitatis Tartuensis de Mathematica. ISSN 1406-2283. 10, p. 31–39. Full text in Research Archive
• Nygaard, Olav Kristian; Abrahamsen, Trond Arnold & Poldvere, Mert (2006). On weak integrability and boundedness in Banach spaces. Journal of Mathematical Analysis and Applications. ISSN 0022-247X. 314, p. 67–74. Full text in Research Archive Show summary

  Thin and thick sets in normed spaces were defined and studied by M.I. Kadets and V.P. Fonf in 1983. In this paper, we give a new characterization of thick sets in terms of weak integrability of Banach space valued measurable functions. We also characterize thick sets in terms of boundedness of vector measures, and explain how this concept is related to the theory of barrelled spaces.

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• Abrahamsen, Trond Arnold (2021). Almost square Banach spaces and relatives. Full text in Research Archive
• Abrahamsen, Trond Arnold; Langemets, Johann; Lima, Vegard & Nygaard, Olav Kristian (2015). Correction to the paper On thickness and thinness in Banach spaces, Houston J. Math., 41 (2015), no. 1, 97-111. Houston Journal of Mathematics. ISSN 0362-1588. 41(4), p. 1359–1360. Full text in Research Archive
• Abrahamsen, Trond Arnold (2015). Diameter 2 properties in Banach spaces. Full text in Research Archive
• Abrahamsen, Trond Arnold (2014). A renorming result for Banach spaces containing c_0. Full text in Research Archive
• Abrahamsen, Trond Arnold (2013). Uniformly square Banach spaces. Full text in Research Archive
• Abrahamsen, Trond Arnold (2013). Uniformly square Banach spaces. Full text in Research Archive
• Abrahamsen, Trond Arnold (2012). Diameter 2 properties in Banach spaces. Part II: Passing to subspaces. Full text in Research Archive
• Abrahamsen, Trond Arnold (2003). A test for bounded semivariation. Full text in Research Archive
• Martiny, Andre; Abrahamsen, Trond Arnold; Lima, Vegard & Dovland, Olav Kristian (2021). The big slice phenomenon in Banach spaces. Universitetet i Agder. ISSN 9788284270418. Full text in Research Archive

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