A strongly diagonal power of algebraic order bounded disjointness preserving operators
Authors:
Karim Boulabiar, Gerard Buskes and Gleb Sirotkin
Journal:
Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 94-98
MSC (2000):
Primary 47B65, 06F20, 06F25
DOI:
https://doi.org/10.1090/S1079-6762-03-00116-1
Published electronically:
October 8, 2003
MathSciNet review:
2029470
Full-text PDF Free Access
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Additional Information
Abstract: An order bounded disjointness preserving operator $T$ on an Archimedean vector lattice is algebraic if and only if the restriction of $T^{n!}$ to the vector sublattice generated by the range of $T^{m}$ is strongly diagonal, where $n$ is the degree of the minimal polynomial of $T$ and $m$ is its ‘valuation’.
- Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory, Graduate Studies in Mathematics, vol. 50, American Mathematical Society, Providence, RI, 2002. MR 1921782
- Y. A. Abramovich and A. K. Kitover, Inverses of disjointness preserving operators, Mem. Amer. Math. Soc. 143 (2000), no. 679, viii+162. MR 1639940, DOI https://doi.org/10.1090/memo/0679
- Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Pure and Applied Mathematics, vol. 119, Academic Press, Inc., Orlando, FL, 1985. MR 809372
- J. Araujo, E. Beckenstein, and L. Narici, Biseparating maps and homeomorphic real-compactifications, J. Math. Anal. Appl. 192 (1995), no. 1, 258–265. MR 1329423, DOI https://doi.org/10.1006/jmaa.1995.1170
- Wolfgang Arendt, Spectral properties of Lamperti operators, Indiana Univ. Math. J. 32 (1983), no. 2, 199–215. MR 690185, DOI https://doi.org/10.1512/iumj.1983.32.32018
- W. Arendt and D. R. Hart, The spectrum of quasi-invertible disjointness preserving operators, J. Funct. Anal. 68 (1986), no. 2, 149–167. MR 852658, DOI https://doi.org/10.1016/0022-1236%2886%2990003-0
BBK. Boulabiar, G. Buskes, Polar decomposition of order bounded disjointness preserving operators, Proc. Amer. Math. Soc., To appear.
BBHK. Boulabiar, G. Buskes, M. Henriksen, A generalization of a theorem on biseparating maps, J. Math. Ana. Appl. 280 (2003), 336–351.
- J. J. Grobler and C. B. Huijsmans, Disjointness preserving operators on complex Riesz spaces, Positivity 1 (1997), no. 2, 155–164. MR 1658332, DOI https://doi.org/10.1023/A%3A1009746711470
- Krzysztof Jarosz, Automatic continuity of separating linear isomorphisms, Canad. Math. Bull. 33 (1990), no. 2, 139–144. MR 1060366, DOI https://doi.org/10.4153/CMB-1990-024-2
KI. Kaplansky, Infinite Abelian Groups, University of Michigan Press, Ann Arbor, 1954.
- W. A. J. Luxemburg, B. de Pagter, and A. R. Schep, Diagonals of the powers of an operator on a Banach lattice, Operator theory in function spaces and Banach lattices, Oper. Theory Adv. Appl., vol. 75, Birkhäuser, Basel, 1995, pp. 223–273. MR 1322506
- Mathieu Meyer, Les homomorphismes d’espaces vectoriels réticulés complexes, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 17, 793–796 (French, with English summary). MR 622421
- Peter Meyer-Nieberg, Banach lattices, Universitext, Springer-Verlag, Berlin, 1991. MR 1128093
AAYu. A. Abramovich, C. D. Aliprantis, An Invitation to Operator Theory, Graduate Studies Math., Vol. 50, Amer. Math. Soc., Providence, RI, 2002.
AKYu. A. Abramovich, A. K. Kitover, Inverses of disjointness preserving operators, Memoirs Amer. Math. Soc. 143 (2000), no. 679.
ABC. D. Aliprantis, O. Burkinshaw, Positive Operators, Academic Press, Orlando, FL, 1985.
ABNJ. Araujo, E. Beckenstein, L. Narici, Biseparating maps and homeomorphic real-compactifications, J. Math. Ana. Appl. 12 (1995), 258–265.
AW. Arendt, Spectral properties of Lamperti operators, Indiana Univ. Math. J. 32 (1983), 199–215.
AHW. Arendt, D. R. Hart, The spectrum of quasi-invertible disjointness preserving operators, J. Funct. Anal. 33 (1986), 149–167.
BBK. Boulabiar, G. Buskes, Polar decomposition of order bounded disjointness preserving operators, Proc. Amer. Math. Soc., To appear.
BBHK. Boulabiar, G. Buskes, M. Henriksen, A generalization of a theorem on biseparating maps, J. Math. Ana. Appl. 280 (2003), 336–351.
GHJ. J. Grobler, C. B. Huijsmans, Disjointness preserving operators on complex Riesz spaces, Positivity 1 (1997), 155–164.
JK. Jarosz, Automatic continuity of separating linear isomorphisms, Bull. Canadian Math. Soc. 33 (1990), 139–144.
KI. Kaplansky, Infinite Abelian Groups, University of Michigan Press, Ann Arbor, 1954.
LPSW. A. J. Luxemburg, B. de Pagter, A. R. Schep, Diagonals of the powers of an operator on a Banach lattice, Operator theory in function spaces and Banach lattices, Oper. Theory Adv. Appl., Vol. 75, Birkhäuser, 1995, pp. 223–273.
MM. Meyer, Les homomorphismes d’espaces vectoriels réticulés complexes, C. R. Acad. Sci. Paris, Serie. I 292 (1981), 793–796.
M-NP. Meyer-Nieberg, Banach Lattices, Springer-Verlag, Berlin, 1991.
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Additional Information
Karim Boulabiar
Affiliation:
IPEST, Université de Carthage, BP 51, 2070-La Marsa, Tunisia
Email:
karim.boulabiar@ipest.rnu.tn
Gerard Buskes
Affiliation:
Department of Mathematics, University of Mississippi, MS 38677
Email:
mmbuskes@olemiss.edu
Gleb Sirotkin
Affiliation:
Department of Mathematics, Northern Illinois University, DeKalb, IL 60115
Email:
sirotkin@math.niu.edu
Keywords:
Algebraic,
disjointness preserving,
locally algebraic,
minimal polynomial,
orthomorphism,
strongly diagonal
Received by editor(s):
June 18, 2003
Published electronically:
October 8, 2003
Additional Notes:
The first and the second authors gratefully acknowledge support from the NATO Collaborative Linkage Grant #PST.CLG.979398. The second author also acknowledges support from the Office of Naval Research Grant #N00014-01-1-0322
Communicated by:
Svetlana Katok
Article copyright:
© Copyright 2003
American Mathematical Society
