Partial isometries and the conjecture of C.K. Fong and S.K. Tsui

We investigate some bounded linear operators T   on a Hilbert space which satisfy the condition |T|≤|ReT||T|≤|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.