Commutator and self-commutator approximants II
Keyword(s):
We minimize the quantities (i) ||T -(AX-X A) ||, (ii) ||T -(X*X-XX*)|| and (iii) ||T -(AX - X B)|| 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 = T B and T A = 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 ||An|| = ||A||n if n?N.