Two problems in the theory of disjointness preserving operators

Year 2021, Volume: 50 Issue: 2, 361 - 364, 11.04.2021

Bahri Turan , Kazim Özcan

https://doi.org/10.15672/hujms.622718

https://izlik.org/JA84PN63GY

Abstract

In this short note, our aim is to solve two problems in the theory of disjointness preserving operators. Firstly, we obtain the converse direction of Hart's Theorem which was given in [D.R. Hart, Some properties of disjointness preserving operators, Mathematics Proceedings, 1985]. As a result, we get an affirmative solution of an open problem given by Y.A. Abramovich and A.K. Kitover in [Inverses of disjointness preserving operators, Mem. Amer. Math. Soc., 2000].

Keywords

Riesz space , disjointness preserving operator , Hart's Theorem

References

• [1] Y.A. Abramovich and A.K. Kitover, Inverses of disjointness preserving operators, Mem. Amer. Math. Soc. 143 (679), 2000.
• [2] C.D. Aliprantis and O. Burkinshaw, Positive Operators, Academic Press, London,1985.
• [3] D.R. Hart, Some properties of disjointness preserving operators, Mathematics Proceedings A 88 (2), 183-197, 1985.
• [4] W.A.J. Luxemburg and A.C. Zaanen, Riesz Space I, North Holland, Amsterdam, 1971.
• [5] A.G. Rugy, La structure ideale des M-espaces, J. Math. Pures at Appl. 51, 331-373, 1972.
• [6] H.H. Schaefer, Banach lattices and positive operators, Springer, Berlin, 1991.
• [7] B. Turan, On ideal operators, Positivity 7, 141-148, 2003.
• [8] A.C. Zaanen, Riesz spaces II, North-Holland, Amsterdam, 1983.