Note on some operator equations and local spectral properties

Il Ju An, Eungil Ko

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


In this paper we define Sk, j by the set of solutions (A,B) of the operator equations AkBj+1Ak = A2k+j and BkAj+1Bk = B2k+j. Then we observe the set Sk, j is increasing for all integers k ≥ 1 and j ≥ 0. Now let a pair (A,B) ∈ Sk, j ∩ Sj+1,k−1 for any integer k ≥ 1 and j ≥ 0. We show that if any one of the operators A, AB, BA, and B has Bishop’s property (β), then all others have the same property. Furthermore, we prove that the operators Ak+j, AkBj+1, Aj+1Bk, Bj+1Ak, BkAj+1 and Bk+j have the same spectra and spectral properties. Finally, we investigate their Weyl type theorems.

Original languageEnglish
Pages (from-to)397-417
Number of pages21
JournalOperators and Matrices
Issue number2
StatePublished - Jun 2016


  • Operator equations
  • Single valued extension property
  • Spectrum


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