Mathematica Bohemica, Vol. 143, No. 2, pp. 201-212, 2018
Some equivalent metrics for bounded normal operators
Mohammad Reza Jabbarzadeh, Rana Hajipouri
Received December 5, 2016. First published October 16, 2017.
Abstract: Some stronger and equivalent metrics are defined on $\mathcal{M}$, the set of all bounded normal operators on a Hilbert space $H$ and then some topological properties of $\mathcal{M}$ are investigated.
Keywords: Hilbert space; normal operator; equivalent metrics; composition operator
References: [1] M. Benharrat, B. Messirdi: Strong metrizability for closed operators and the semi-Fredholm operators between two Hilbert spaces. Int. J. Anal. Appl. 8 (2015), 110-122. [2] H. O. Cordes, J. P. Labrousse: The invariance of the index in the metric space of closed operators. J. Math. Mech. 12 (1963), 693-719. MR 0162142 | Zbl 0148.12402 [3] F. Gilfeather: Norm conditions on resolvents of similarities of Hilbert space operators and applications to direct sums and integrals of operators. Proc. Am. Math. Soc. 68 (1978), 44-48. DOI 10.2307/2040905 | MR 0475583 | Zbl 0381.47003 [4] T. Kato: Perturbation Theory for Linear Operators. Grundlehren der mathematischen Wissenschaften 132. Springer, Berlin (1976). DOI 10.1007/978-3-642-66282-9 | MR 0407617 | Zbl 0342.47009 [5] W. E. Kaufman: A stronger metric for closed operators in Hilbert space. Proc. Am. Math. Soc. 90 (1984), 83-87. DOI 10.2307/2044673 | MR 0722420 | Zbl 0551.47001 [6] F. Kittaneh: On some equivalent metrics for bounded operators on Hilbert space. Proc. Am. Math. Soc. 110 (1990), 789-798. DOI 10.2307/2047922 | MR 1027097 | Zbl 0721.47013 [7] J. P. Labrousse: On a metric space of closed operators on a Hilbert space. Univ. Nac. Tucumán, Rev., Ser. A 16 (1966), 45-77. MR 0226445 | Zbl 0154.15803 [8] J. P. Labrousse: Quelques topologies sur des espaces d'opérateurs dans des espaces de Hilbert et leurs applications. Faculté des Sciences de Nice (Math.) 1 (1970), 47 pages. [9] A. Lambert, S. Petrovic: Beyond hyperinvariance for compact operators. J. Funct. Anal. 219 (2005), 93-108. DOI 10.1016/j.jfa.2004.06.001 | MR 2108360 | Zbl 1061.47018 [10] R. K. Singh, J. S. Manhas: Composition Operators on Function Spaces. North-Holland Mathematics Studies 179. North-Holland, Amsterdam (1993). MR 1246562 | Zbl 0788.47021
Affiliations: Mohammad Reza Jabbarzadeh, Rana Hajipouri, Faculty of Mathematical Sciences, University of Tabriz, P. O. Box: 5166616471, Tabriz, Iran, e-mail: mjabbar@tabrizu.ac.ir, r.hajipouri@tabrizu.ac.ir
