NOTE ON SOME OPERATOR EQUATIONS AND LOCAL SPECTRAL PROPERTIES

Abstract

In this paper we define J(k, j) by the set of solutions (A, B) of the operator equations A(k)B(j+1)A(k) = A(2k+j) and B(k)A(j+1)B(k) = B2k+j. Then we observe the set J(k,j) is increasing for all integers k >= 1 and j >= 0. Now let a pair (A, B) is an element of J(k,j) boolean AND J(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 (beta), then all others have the same property. Furthermore, we prove that the operators A(k+j), A(k)B(j+1), A(j+1)B(k), B(j+1)A(k), B(k)A(j+1) and Bk+j have the same spectra and spectral properties. Finally, we investigate their Weyl type theorems.