The Hilbert–Schmidt version of the commutator theorem for zero trace matrices. (1st April 2015) - Angel, Omer; Schechtman, Gideon
Abstract : LetA be anm × m complex matrix with zero trace. Then there arem × m matricesB andC such thatA = [ B, C ] and∥ B ∥ ∥ C ∥ 2 ⩽ ( log m + O ( 1 ) ) 1 / 2 ∥ A ∥ 2 where∥ D ∥ is the norm ofD as an operator onℓ 2 m and∥ D ∥ 2 is the Hilbert–Schmidt norm ofD . Moreover, the matrixB can be taken to be normal. Conversely, there is a zero tracem × m matrixA such that wheneverA = [ B, C ], ∥ B ∥ ∥ C ∥ 2 ⩾ | log m − O ( 1 ) | 1 / 2 ∥ A ∥ 2 for some absolute constantc > 0 .