Let (BN) denote the class
of all bounded linear operators on a Hilbert space such that τ ∗τ and TT∗ commute.
Let (BN)+ be those τ ∈ (BN) which are hyponorma]. Embry has observed that
if T ∈ (BN), then 0 ∈ W(T) or T is normal. This is used to show that if
τ ∈ (BN), then (T + λI)∉(BN) unless T is normal. It is also shown that if
τ ∈ (BN)+, then Tn is hyponormal for n ≧ 1. An example of a T ∈ (BN)+ such
that T2∉(BN) is given. Paranormality of operators in (BN) is shown to be
equivalent to hyponormality. The relationship between T being in (BN)
and T being centered is discussed. Finally, all 3 ×3 matrices in (BN) are
characterized.
