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We prove an analogue to the Cayley identity for an arbitrary self-adjointoperator in a Hilbert space. We also provide two new ways to characterizevectors belonging to the singular spectral subspace in terms of the analyticproperties of the resolvent of the operator, computed on these vectors. Thelatter are analogous to those used routinely in the scattering theory for theabsolutely continuous subspace.

On Cayley Identity for Self-Adjoint Operators in Hilbert Spaces