## Annals of Functional Analysis

### On the modulus of disjointness-preserving operators and $b$-$AM$-compact operators on Banach lattices

#### Abstract

We study several properties of the modulus of order bounded disjointness-preserving operators. We show that, if $T$ is an order bounded disjointness-preserving operator, then $T$ and $\vert T\vert$ have the same compactness property for several types of compactness. Finally, we characterize Banach lattices having $b$-$\mathit{AM}$-compact (resp., $\mathit{AM}$-compact) operators defined between them as having a modulus that is $b$-$\mathit{AM}$-compact (resp., $\mathit{AM}$-compact).

#### Article information

Source
Ann. Funct. Anal., Volume 9, Number 1 (2018), 101-110.

Dates
Accepted: 27 February 2017
First available in Project Euclid: 14 August 2017

https://projecteuclid.org/euclid.afa/1502697620

Digital Object Identifier
doi:10.1215/20088752-2017-0027

Mathematical Reviews number (MathSciNet)
MR3758746

Zentralblatt MATH identifier
06841344

Subjects
Secondary: 47B60: Operators on ordered spaces

#### Citation

Haghnezhad Azar, Kazem; Alavizadeh, Razi. On the modulus of disjointness-preserving operators and $b$ - $AM$ -compact operators on Banach lattices. Ann. Funct. Anal. 9 (2018), no. 1, 101--110. doi:10.1215/20088752-2017-0027. https://projecteuclid.org/euclid.afa/1502697620

#### References

• [1] C. D. Aliprantis and O. Burkinshaw, Positive Operators, Springer, Dordrecht, 2006.
• [2] Ş. Alpay and B. Altın, A note on $b$-weakly compact operators, Positivity 11 (2007), no. 4, 575–582.
• [3] B. Aqzzouz and J. H’michane, Some results on order weakly compact operators, Math. Bohem. 134 (2009), no. 4, 359–367.
• [4] B. Aqzzouz and J. H’michane, The class of $b$-$AM$-compact operators, Quaest. Math. 36 (2013), no. 3, 309–319.
• [5] B. Aqzzouz, R. Nouira, and L. Zraoula, Compactness properties of operators dominated by $AM$-compact operators, Proc. Amer. Math. Soc. 135 (2007), no. 4, 1151–1157. Correction, Proc. Amer. Math. Soc. 137 (2009), no. 8, 2813-2815.
• [6] W. Arendt, Spectral properties of Lamperti operators, Indiana Univ. Math. J. 32 (1983), no. 2, 199–215.
• [7] K. Boulabiar, Recent trends on order bounded disjointness preserving operators, Irish Math. Soc. Bull. 62 (2008), 43–69.
• [8] K. Boulabiar and G. Buskes, Polar decomposition of order bounded disjointness preserving operators, Proc. Amer. Math. Soc. 132 (2004), no. 3, 799–806.
• [9] A. E. Kaddouri, J. Hmichane, K. Bouras, and M. Moussa, Some results on $b$-$AM$-compact operators, Complex Anal. Oper. Theory 7 (2013), no. 6, 1889–1895.
• [10] A. G. Kusraev and S. S. Kutateladze, On order bounded disjointness preserving operators, Sib. Math. J. 55 (2014), no. 5, 915–928.
• [11] P. Meyer-Nieberg, Banach Lattices, Springer, Berlin, 2012.
• [12] X. D. Zhang, Decomposition theorems for disjointness preserving operators, J. Funct. Anal. 116 (1993), no. 1, 158–178.