Abstract
We show that a semigroup of positive matrices (all entries greater than or equal to zero) with binary diagonals (diagonal entries either 0 or 1) is either decomposable (all matrices in the semigroup have a common zero entry) or is similar, via a positive diagonal matrix, to a binary semigroup (all entries 0 or 1). In the case where the idempotents of minimal rank in \({\mathcal{S}}\) satisfy a “diagonal disjointness” condition, we obtain additional structural information. In the case where the semigroup is not necessarily positive but has binary diagonals we show that either the semigroup is reducible or the minimal rank ideal is a binary semigroup. We also give generalizations of these results to operators acting on the Hilbert space of square-summable sequences.
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References
Berman M., Plemmons R.: Nonnegative Matrices in Mathematical Sciences. Academic Press, New York (1979)
Semigroups Working Group at LAW ’08, Kranjska Gora, Semigroups of operators with nonnegative diagonals. Lin. Alg. Appl. (2010 to appear)
Bernik J., Mastnak M., Radjavi H.: Realizing irreducible semigroups and real algebras of compact operators. J. Math. Anal. Appl. 348(2), 692–707 (2008)
Bernik, J., Mastnak, M., Radjavi, H.: Positivity and matrix semigroups. Lin. Alg. Appl. (2010 to appear)
Burnside W.: On the condition of reducibility of any group of linear substitutions. Proc. Lond. Math. Soc. 3, 430–434 (1905)
Choi M.D., Nordgren E.A., Radjavi H., Rosenthal P., Zhong Y.: Triangularizing semigroups of quasinilpotent operators with nonnegative entries. Indiana Univ. Math. J. 42, 15–25 (1993)
Drnovsek R.: Common invariant subspaces for collections of operators. Intgral Equ. Oper. Theory 39, 253–266 (2001)
Gessesse H., Popov A., Radjavi H., Spinu E., Tcaciuc A., Troitsky V.: Bounded indecomposable semigroups of non-negative matrices. Positivity 14(3), 383–394 (2010)
Jahandideh M.T.: On the ideal-triangularizability of positive operators on Banach lattices. Proc. Am. Math. Soc. 125, 2661–2670 (1997)
Kaplansky I.: The Engel-Kolchin theorem revisited. In: Bass, H., Cassidy, P.J., Kovaciks, J. (eds) Contributions to Algebra, pp. 233–237. Academic Press, New York (1977)
Radjavi H., Rosenthal P.: Simultaneous Triangularization, Universitext. Springer, Berlin (2000)
Radjavi H., Rosenthal P.: Limitations on the size of semigroups of matrices. Semigroup Forum 76, 25–31 (2008)
Ringrose J.R.: Super-diagonal forms for compact linear operators. Proc. Lond. Math. Soc. 12(3), 367–384 (1962)
Zhong Y.: Functional positivity and invariant subspaces of operators. Houst. J. Math. 19, 239–262 (1993)
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G. MacDonald and H. Radjavi acknowledge the support of NSERC Canada.
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Livshits, L., MacDonald, G. & Radjavi, H. Positive matrix semigroups with binary diagonals. Positivity 15, 411–440 (2011). https://doi.org/10.1007/s11117-010-0092-6
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DOI: https://doi.org/10.1007/s11117-010-0092-6