Trond Arnold Abrahamsen

Professor

Department of Mathematical Sciences

Email: trond.a.abrahamsen@uia.no
Phone: +47 38141773
Mobile phone: +4747293249

Office J2021 (Universitetsveien 25, 4630 Kristiansand, Norway)

Research groups

Pure and Applied Mathematical Analysis Research group (PAMAR)

Areas of responsibility

Educational background

M.Sc. Mathematics Education, Department of Mathematical Sciences, Agder University College, 2001.

Ph.D. Dr. Scient in mathematics (Functional analysis), Department of Mathematics University of Bergen, 2007.

Teaching quaifications

Postgraduate Certificate in Education, Agder University College, 1999.

Professional experience

2001 - 2002: Assistent Professor at the Department of Mathematical Sciences, Agder University College.

2002 - 2006: PhD student at the Department Mathematics, University of Bergen.

2006 - 2007: Assistent Professor at the Department of Mathematical Sciences, Agder University College.

2007 - 2021: Associate Professor at the Department of Mathematical Sciences, University of Agder.

2021 - current day: Professor at the Department of Mathematical Sciences, University of Agder.

Research

Academic interests

Geometry of Banach spaces.

Are teaching a large selection of the courses in mathematics at the department.

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 (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.
Abrahamsen, Trond Arnold; Lima, Vegard; Martiny, Andre & Perreau, Yoël (2022). Asymptotic geometry and Delta-points. Banach Journal of Mathematical Analysis. ISSN 2662-2033. 16. 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 & Troyanski, Stanimir (2021). Daugavet- and delta-points in Banach spaces with unconditional bases. Transactions of the American Mathematical Society - Series B. ISSN 2330-0000. 8(13), p. 379–398. doi: 10.1090/btran/68. 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.
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; Becerra Guerrero, Julio; 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; Dovland, Olav & Poldvere, Mert (2019). New applications of extremely regular function spaces. Pacific Journal of Mathematics. ISSN 0030-8730. doi: 10.2140/pjm.2019.301.385. Full text in Research Archive
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.
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
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.
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.
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.
Abrahamsen, Trond Arnold (2015). Linear extensions, almost isometries, and diameter two. Extracta Mathematicae. ISSN 0213-8743. 30(2), p. 135–151.
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.
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.
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.
Abrahamsen, Trond Arnold; Lima, Vegard & Dovland, Olav (2013). Remarks on diameter 2 properties. Journal of Convex Analysis. ISSN 0944-6532.
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.
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.
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.
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. 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.
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.
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. 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.

View all works in Cristin

Abrahamsen, Trond Arnold (2021). Almost square Banach spaces and relatives.
Abrahamsen, Trond Arnold (2015). Diameter 2 properties in Banach spaces.
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.
Abrahamsen, Trond Arnold (2014). A renorming result for Banach spaces containing c_0.
Abrahamsen, Trond Arnold (2013). Uniformly square Banach spaces.
Abrahamsen, Trond Arnold (2013). Uniformly square Banach spaces.
Abrahamsen, Trond Arnold (2012). Diameter 2 properties in Banach spaces. Part II: Passing to subspaces.
Abrahamsen, Trond Arnold (2012). Matematikkstudier ved UiA - Til mulig student hos oss (Vennesla vgs).
Abrahamsen, Trond Arnold (2012). Matematikkstudier ved UiA - Til mulig student hos oss (Kristiansand Katedralskole).
Abrahamsen, Trond Arnold (2003). A test for bounded semivariation.
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

View all works in Cristin

Published Apr. 16, 2024 11:26 AM - Last modified June 13, 2024 12:55 PM