Commutator, and self-commutator, approximants, 2

We minimize the quantities (i) kT ¡ (AX ¡ XA)k, (ii) kT ¡ (X¤X ¡
XX¤)k and (iii) kT ¡ (AX ¡ XB)k where T is isometric and where in (i)
A is paranormal and commutes with T, in (ii) X¤ (or X) is paranormal
and commutes with T, and in (iii) A and B are paranormal and AT = TB
and TA = BT. The upshot is that these quantities are minimized when
0 = AX ¡XA = X¤X ¡XX¤ = AX ¡XB. To prove these results we obtain
the power norm equality for paranormal operators: if A is paranormal then
kAnk = kAkn if n 2 N.