Abstract
For the lower Weyl spectrum
where T is a positive operator on a Banach lattice E, the conditions for which the equality \({\sigma_{\rm w}^-(T) = \sigma_{\rm w}^-(T^*)}\) holds, are established. In particular, it is true if E has order continuous norm. An example of a weakly compact positive operator T on ℓ ∞ such that the spectral radius \({r(T) \in \sigma_{\rm w}^-(T) {\setminus} (\sigma_{\rm f}(T) \cup \sigma_{\rm w}^-(T^*))}\) , where σ f(T) is the Fredholm spectrum, is given. The conditions which guarantee the order continuity of the residue T −1 of the resolvent R(., T) of an order continuous operator T ≥ 0 at \({r(T) \notin \sigma_{\rm f}(T)}\) , are discussed. For example, it is true if T is o-weakly compact. It follows from the proven results that a Banach lattice E admitting an order continuous operator T ≥ 0, \({r(T) \notin \sigma_{\rm f}(T)}\) , can not have the trivial band \({E_n^\sim}\) of order continuous functionals in general. It is obtained that a non-zero order continuous operator T : E → F can not be approximated in the r-norm by the operators from \({E_\sigma^\sim \otimes F}\) , where F is a Banach lattice, \({E_\sigma^\sim}\) is a disjoint complement of the band \({E_n^\sim}\) of E*.
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 (2002)
Abramovich Y.A., Sirotkin G.: On order convergence of nets. Positivity 9(3), 287–292 (2005)
Aiena P.: Fredholm and Local Spectral Theory, with Applications to Multipliers. Kluwer, Dordrecht (2004)
Alekhno E.A.: Some special properties of Mazurs’ functionals, I. Trans. Math. Inst. Nats. Akad. Navuk Belarusi 12(1), 17–20 (2004) (Russian)
Alekhno, E.A. Some properties of the weak topology in the space \({\mathcal{L}_\infty}\) . Vestsi Nats. Akad. Navuk Belarusi. Ser. Fiz.-Mat. Navuk (3), 31–37 (2006) (Russian)
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)
Aliprantis C.D., Burkinshaw O.: Positive Operators. Academic Press, New York (1985)
Arendt W.: On the o-spectrum of regular operators and the spectrum of measures. Math. Z. 178(2), 271–287 (1981)
Caselles V.: On the peripheral spectrum of positive operators. Isr. J. Math. 58(2), 144–160 (1987)
Dunford N., Schwartz J.T.: Linear Operators Part 1: General Theory. Wiley, New York (1958)
Grobler J.J., Reinecke C.J.: On principal T-bands in a Banach lattice. Integral Equ. Oper. Theory 28(4), 444–465 (1997)
Kitover A.K., Wickstead A.W.: Operator norm limits of order continuous operators. Positivity 9(2), 341–355 (2005)
Lozanovsky G.Y.: On projections on some Banach lattices. Mat. Zametki 4(1), 41–44 (1968) (Russian)
Meyer-Nieberg P.: Banach Lattices. Springer-Verlag, Berlin (1991)
Schaefer H.H.: Banach Lattices and Positive Operators. Springer-Verlag, Berlin (1974)
Wickstead A.W.: Converses for the Dodds-Fremlin and Kalton-Saab theorems. Math. Proc. Cambr. Philos. Soc. 120(1), 175–179 (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Alekhno, E.A. The Lower Weyl Spectrum of a Positive Operator. Integr. Equ. Oper. Theory 67, 301–326 (2010). https://doi.org/10.1007/s00020-010-1758-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00020-010-1758-y