A Contribution to the Fong-Tsui Conjecture Related to Self-adjoint   Operators
  We are interested in an open question raised by Fong-Tsui (dating back to the beginning of the eighties of last century) as to whether a bounded operator whose absolute value is less than the absolute value of its real part is self-adjoint. The analogue in the unbounded operators setting is also treated. 
Mohammed Hichem Mortad
math.FA