Note on some operator equations and local spectral properties

In this paper we define S_{k, j} by the set of solutions (A,B) of the operator equations A^kB^j+1A^k = A^2k+j and B^kA^j+1B^k = B^2k+j. Then we observe the set S_{k, j} is increasing for all integers k ≥ 1 and j ≥ 0. Now let a pair (A,B) ∈ S_{k, 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 A^k+j, A^kB^j+1, A^j+1B^k, B^j+1A^k, B^kA^j+1 and B^k+j have the same spectra and spectral properties. Finally, we investigate their Weyl type theorems.