# zbMATH — the first resource for mathematics

Finite spectra and quasinilpotent equivalence in Banach algebras. (English) Zbl 1274.46094
The notion of quasinilpotent equivalence was introduced by I. Colojoară and C. Foiaş [Theory of generalized spectral operators. Mathematics and its Applications. 9. New York-London-Paris: Gordon and Breach Science Publishers (1968; Zbl 0189.44201)].
It is well known that the quasinilpotent equivalence preserves the spectrum. The present paper shows that quasinilpotent equivalent elements with finite spectrum in a semisimple Banach algebra even have equal Riesz projections. In particular, quasinilpotent equivalent elements in the socle have the same spectral multiplicities, traces and determinants.
The paper further studies the quasinilpotent equivalence for normal elements in $$C^{*}$$-algebras with finite or infinite spectra.
Reviewer: V. Müller (Praha)

##### MSC:
 46H05 General theory of topological algebras
##### Keywords:
quasinilpotent equivalence; finite rank element
Full Text:
##### References:
 [1] B. Aupetit: A Primer on Spectral Theory. Springer, New York, 1991. [2] B. Aupetit, H. du T. Mouton: Trace and determinant in Banach algebras. Stud. Math. 121 (1996), 115–136. · Zbl 0872.46028 [3] F. F. Bonsall, J. Duncan: Complete Normed Algebras. Springer, New York, 1973. · Zbl 0271.46039 [4] R. Brits: Perturbation and spectral discontinuity in Banach algebras. Stud. Math. 203 (2011), 253–263. · Zbl 1254.46049 [5] I. Colojoarǎ, C. Foiaş: Quasi-nilpotent equivalence of not necessarily commuting operators. J. Math. Mech. 15 (1966), 521–540. · Zbl 0138.07701 [6] I. Colojoarǎ, C. Foiaş: Theory of Generalized Spectral Operators. Mathematics and its Applications. 9, New York-London-Paris: Gordon and Breach Science Publishers, 1968. [7] C. Foiaş, F.-H. Vasilescu: On the spectral theory of commutators. J. Math. Anal. Appl. 31 (1970), 473–486. · Zbl 0175.13604 [8] R. Harte: On rank one elements. Stud. Math. 117 (1995), 73–77. · Zbl 0837.46036 [9] S. Mouton, H. Raubenheimer: More spectral theory in ordered Banach algebras. Positivity 1 (1997), 305–317. · Zbl 0904.46036 [10] V. Müller: Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras. Operator Theory: Advances and Applications, Basel: Birkhäuser, 2003. [11] J. Puhl: The trace of finite and nuclear elements in Banach algebras. Czech. Math. J. 28 (1978), 656–676. · Zbl 0394.46041 [12] H. Raubenheimer: On quasinilpotent equivalence in Banach algebras. Czech. Math. J. 60 (2010), 589–596. · Zbl 1224.46091 [13] M. Razpet: The quasinilpotent equivalence in Banach algebras.. J. Math. Anal. Appl. 166 (1992), 378–385. · Zbl 0802.46064
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.