The Hilbert–Schmidt version of the commutator theorem for zero trace matrices

angel, omer; schechtman, gideon;

Let A be an m × m complex matrix with zero trace. Then there are m × m matrices B and C such that A = [B,C] and |B||C|2 <= (logm + O(1))1/2|A|2 where |D| is the norm of D as an operator on m2
and |D|2 is the Hilbert–Schmidt norm of D. Moreover, the matrix B can be taken to be normal. Conversely, there is a zero trace m × m matrix A such that whenever A = [B,C], |B||C|2 >= | logm − O(1)|1/2|A|2 for some absolute constant c > 0.