A Spectral Representation for Bounded Non-Selfadjoint Operators

An integral representation for an arbitrary bounded operator T defined on a Hilbert space ${mathcal{H}}$ is given. The representing measure is in general defined on a Jordan curve surrounding the spectrum of T. It is obtained as a limit, in a certain weak sense, of a family (F _r ) of absolutely continuous measures the Radon?CNikodym derivative of which (with respect to the standard Lebesgue measure on the considered Jordan curve) are described explicitly in terms of the operator T and its adjoint T ^*.