Abstract
One of the central problems investigated in Alekhno (Positivity 11(3):375–386, 2007, Positivity 13(1):3–20, 2009) is that of providing conditions under which the spectral radius of a positive operator T on a complex Banach lattice lies outside the lower Weyl spectrum of T given that it is not an element of its essential spectrum. In this paper the lower Weyl spectrum of an arbitrary positive ordered Banach algebra element is introduced and studied, and work done in the aforementioned papers are extended to general ordered Banach algebras.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramovich, Y.A., Aliprantis, C.D.: An invitation to operator theory. In: Graduate Studies in Mathematics, vol. 50. American Mathematical Society, Providence (2002)
Alekhno, E.A.: Some properties of essential spectra of a positive operator. Positivity 11(3), 375–386 (2007)
Alekhno, E.A.: Some properties of essential spectra of a positive operator, II. Positivity 13(1), 3–20 (2009)
Alekhno, E.A.: The lower Weyl spectrum of a positive operator. Integr. Equ. Oper. Theory 67(3), 301–326 (2010)
Alekhno, E.A.: The irreducibility in ordered Banach algebras. Positivity 16(1), 143–176 (2012)
Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Academic Press, Orlando (1985)
Arendt, W., Sourour, A.R.: Perturbation of regular operators and the order essential spectrum. Indag. Math. 48(2), 109–122 (1986)
Aupetit, B.: A Primer on Spectral Theory. Springer, New York (1991)
Benjamin, R.: Fredholm theory in ordered Banach algebras. Ph.D. thesis, Stellenbosch University. http://hdl.handle.net/10019.1/98608 (2016)
Benjamin, R., Mouton, S.: Fredholm theory in ordered Banach algebras. Quaest. Math. 39(5), 643–664 (2016)
Benjamin, R., Mouton, S.: The upper Browder spectrum property. Positivity 21(2), 575–592 (2017)
Braatvedt, G., Brits, R., Raubenheimer, H.: Gelfand-Hille type theorems in ordered Banach algebras. Positivity 13(1), 39–50 (2009)
Harte, R.E.: The exponential spectrum in Banach algebras. Proc. Am. Math. Soc. 58(1), 114–118 (1976)
Harte, R.E.: Fredholm theory relative to a Banach algebra homomorphism. Math. Z. 179(3), 431–436 (1982)
Martinez, J., Mazón, J.M.: Quasi-compactness of dominated positive operators and \(C_0\)-semigroups. Math. Z. 207(1), 109–120 (1991)
Mouton, H. du T., Mouton, S., Raubenheimer, H.: Ruston elements and Fredholm theory relative to arbitrary homomorphisms. Quaest. Math. 34(3), 341–359 (2011)
Mouton, S., Raubenheimer, H.: More spectral theory in ordered Banach algebras. Positivity 1(4), 305–317 (1997)
Mouton, S (née Rode).: A spectral problem in ordered Banach algebras. Bull. Aust. Math. Soc. 67(1), 131–144 (2003)
Raubenheimer, H., Rode, S.: Cones in Banach algebras. Indag. Math. (N.S.) 7(4), 489–502 (1996)
Schaefer, H.H.: Banach Lattices and Positive Operators. Springer, New York (1974)
Schaefer, H.H.: On the \(o\)-spectrum of order bounded operators. Math. Z. 154(1), 79–84 (1977)
Acknowledgements
The work was supported by the National Research Foundation (NRF) of South Africa (Grant Nos. 84602 and 96130).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Benjamin, R., Mouton, S. A note on the lower Weyl and Lozanovsky spectra of a positive element. Positivity 22, 533–549 (2018). https://doi.org/10.1007/s11117-017-0526-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-017-0526-5