Balanced Operators and Operator Domains.

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.