Uniqueness under spectral variation in the socle of a Banach algebra

F. Schulz, R. Brits

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


Let A be a complex semisimple Banach algebra with identity, and denote by σ(x) and ρ(x) the nonzero spectrum and spectral radius of an element x∈A, respectively. We explore the relationship between elements a,b∈A that satisfy one of the following conditions: (1) σ(ax)⊆σ(bx) for all x∈A, (2) ρ(ax)≤ρ(bx) for all x∈A. The latter problem was identified by Brešar and Špenko in [7]. In particular, we use these conditions to spectrally characterize prime Banach algebras amongst the class of Banach algebras with nonzero socles, as well as to obtain spectral characterizations of socles which are minimal two-sided ideals.

Original languageEnglish
Pages (from-to)1626-1639
Number of pages14
JournalJournal of Mathematical Analysis and Applications
Issue number2
Publication statusPublished - 15 Dec 2016


  • Rank
  • Socle
  • Spectral radius
  • Spectrum
  • Trace

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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