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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 ∩S j+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. Mathematics subject classification (2010): Primary 47A10, 47A53; Secondary 47B20.

Note on some operator equations and local spectral properties