  Closed operators in Hilbert space defined by a non-self-adjoint resolution of
the identity $\{X(\lambda)\}_{\lambda\in {\mb R}}$, whose adjoints constitute
also a resolution of the identity, are studied . In particular, it is shown
that a closed operator $B$ has a spectral representation analogous to the
familiar one for self-adjoint operators if and only if $B=TAT^{-1}$ where $A$
is self-adjoint and $T$ is a bounded operator with bounded inverse.
