  We shall say that a densely defined closed operator $T$ on a Hilbert space is
balanced if $\cD(T)=\cD(T^*)$. Balanced operators are described in terms of
their phase operators abnd their moduli. Examples of balanced operators are
developed. A characterization of the domain equality $\cD(A)=\cD(B)$ for
positive self-adjoint operators $A$ and $B$ with bounded inverses is given in
terms of their spectral measures.
