Complete operator

A generalized wave operator, i.e. a partially isometric operator defined by

where and are self-adjoint operators on a separable Hilbert space , is an ortho-projector into , and such that

Here , , is the set of all elements that are spectrally absolutely continuous with respect to , i.e. for which the spectral measure of a set is absolutely continuous with respect to the Lebesgue measure .

If the operator , or the analogously defined operator , exists and is complete, the (the parts of the operators on ) are unitarily equivalent. If and are self-adjoint operators on and , where and is real, then and exist and are complete.

References

[1]	T. Kato, "Perturbation theory for linear operators" , Springer (1966) pp. Chapt. X Sect. 3

Comments

An ortho-projector is usually called and orthogonal projector in the West.

An operator is partially isometric if there is a closed linear subspace of such that for and for , the orthogonal complement of ; the set is called the initial set of and the final set of .