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Closed operators in Hilbert space defined by a non-self-adjoint resolution ofthe identity $\{X(\lambda)\}_{\lambda\in {\mb R}}$, whose adjoints constitutealso a resolution of the identity, are studied . In particular, it is shownthat a closed operator $B$ has a spectral representation analogous to thefamiliar 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.

Non-self-adjoint resolutions of the identity and associated operators