Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
cone; positive operator; commutator; spectral radius
cone; positive operator; commutator; spectral radius
Summary:
We establish several inequalities for the spectral radius of a positive commutator of positive operators in a Banach space ordered by a normal and generating cone. The main purpose of this paper is to show that in order to prove the quasi-nilpotency of the commutator we do not have to impose any compactness condition on the operators under consideration. In this way we give a partial answer to the open problem posed in the paper by J. Bračič, R. Drnovšek, Y. B. Farforovskaya, E. L. Rabkin, J. Zemánek (2010). Inequalities involving an arbitrary commutator and a generalized commutator are also discussed.
References:
[1] Bračič, J., Drnovšek, R., Farforovskaya, Y. B., Rabkin, E. L., Zemánek, J.: On positive commutators. Positivity 14 (2010), 431-439. DOI 10.1007/s11117-009-0028-1 | MR 2680506 | Zbl 1205.47040
[2] Daneš, J.: On local spectral radius. Čas. Pěst. Mat. 112 (1987), 177-187. MR 0897643 | Zbl 0645.47002
[3] Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985). MR 0787404 | Zbl 0559.47040
[4] Drnovšek, R., Kandić, M.: More on positive commutators. J. Math. Anal. Appl. 373 (2011), 580-584. DOI 10.1016/j.jmaa.2010.07.056 | MR 2720706 | Zbl 1205.47041
[5] Drnovšek, R.: Once more on positive commutators. Stud. Math. 211 (2012), 241-245. DOI 10.4064/sm211-3-5 | MR 3002445 | Zbl 1267.47062
[6] Esajan, A. R.: Estimating the spectrum of sums of positive semi-commuting operators. Sib. Mat. J. 7 374-378 (1966), translation from Sib. Mat. Zh. 7 (1966), 460-464 Russian. MR 0194893
[7] Förster, K.-H., Nagy, B.: On the local spectral theory of positive operators. Special Classes of Linear Operators and Other Topics (Conference on operator theory, Bucharest, 1986) 71-81 Birkhäuser, Basel (1988). MR 0942914 | Zbl 0649.47001
[8] Förster, K.-H., Nagy, B.: On the local spectral radius of a nonnegative element with respect to an irreducible operator. Acta Sci. Math. (Szeged) 55 (1991), 155-166. MR 1124954 | Zbl 0757.47002
[9] Gao, N.: On commuting and semi-commuting positive operators. (to appear) in Proc. Am. Math. Soc., arXiv:1208.3495 [math.FA]. MR 3209328
[10] Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Notes and Reports in Mathematics in Science and Engineering 5 Academic Press, Boston (1988). MR 0959889 | Zbl 0661.47045
[11] Kittaneh, F.: Spectral radius inequalities for Hilbert space operators. Proc. Am. Math. Soc. (electronic) 134 (2006), 385-390. DOI 10.1090/S0002-9939-05-07796-8 | MR 2176006 | Zbl 1081.47010
[12] Kittaneh, F.: Norm inequalities for commutators of self-adjoint operators. Integral Equations Oper. Theory 62 (2008), 129-135. DOI 10.1007/s00020-008-1605-6 | MR 2442906 | Zbl 1195.47008
[13] Kittaneh, F.: Norm inequalities for commutators of positive operators and applications. Math. Z. 258 (2008), 845-849. DOI 10.1007/s00209-007-0201-9 | MR 2369059 | Zbl 1139.47009
[14] Krasnosel'skiĭ, M. A., Vaĭnikko, G. M., Zabreĭko, P. P., Rutitskii, Ya. B., Stetsenko, V. Ya.: Approximate Solution of Operator Equations. Wolters-Noordhoff Series of Monographs and Textbooks on Pure and Applied Mathematics Wolters-Noordhoff, Groningen (1972). MR 0385655
[15] Laursen, K. B., Neumann, M. M.: An Introduction to Local Spectral Theory. London Mathematical Society Monographs. New Series 20 Clarendon Press, Oxford (2000). MR 1747914 | Zbl 0957.47004
[16] Riesz, F., S.-Nagy, B.: Functional Analysis. Dover Publications, New York (1990); Reprint of the 1955 orig. publ. by Ungar Publ. Co. MR 0071727
[17] Schaefer, H.: Some spectral properties of positive linear operators. Pac. J. Math. 10 (1960), 1009-1019. DOI 10.2140/pjm.1960.10.1009 | MR 0115090 | Zbl 0129.08801
[18] Schaefer, H.: Banach Lattices and Positive Operators. Die Grundlehren der mathematischen Wissenschaften. Band 215 Springer, Berlin (1974). MR 0423039 | Zbl 0296.47023
[19] Zima, M.: On the local spectral radius in partially ordered Banach spaces. Czech. Math. J. 49 (1999), 835-841. DOI 10.1023/A:1022413403733 | MR 1746709 | Zbl 1008.47004
[20] Zima, M.: On the local spectral radius of positive operators. Proc. Am. Math. Soc. (electronic) 131 (2003), 845-850. DOI 10.1090/S0002-9939-02-06726-6 | MR 1937422 | Zbl 1055.47006
[21] Zima, M.: Positive Operators in Banach Spaces and Their Applications. Wydawnictwo Uniwersytetu Rzeszowskiego, Rzeszów (2005). MR 2493071 | Zbl 1165.47002
Search
Partner of
