  We investigate some bounded linear operators T on a Hilbert space which
satisfy the condition |T | less or equal to |ReT |. We describe the maximum
invariant subspace for a contraction T on which T is a partial isometry to
obtain that, in certain cases, the above condition ensures that T is
self-adjoint. In other words we show that the Fong-Tsui conjecture holds for
partial isometries, contractive quasi-isometries, or 2-quasi-isometries, and
Brownian isometries of positive covariance, or even for a more general class of
operators.
