Complete operator

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

$$ W _ {+} ( A _ {2} , A _ {1} ) = s - \lim\limits _ {t \rightarrow x } e ^ {it A _ {2} - it A _ {1} } P _ {1} , $$

where $ A _ {1} $ and $ A _ {2} $ are self-adjoint operators on a separable Hilbert space $ H $, $ P _ {1} $ is an ortho-projector into $ H _ {1,ac} $, and such that

$$ \{ {W _ {+} ( A _ {2} , A _ {1} ) x } : { \| W _ {+} ( A _ {2} , A _ {1} ) x \| = \| x \| } \} = \ H _ {2,ac} . $$

Here $ H _ {i,ac} $, $ i = 1, 2 $, is the set of all elements $ x $ that are spectrally absolutely continuous with respect to $ A _ {i} $, i.e. for which the spectral measure $ \langle E _ {A _ {i} } ( \mu ) x, x \rangle $ of a set $ M $ is absolutely continuous with respect to the Lebesgue measure $ \mu $.

If the operator $ W _ {+} ( A _ {2} , A _ {1} ) $, or the analogously defined operator $ W _ {-} ( A _ {2} , A _ {1} ) $, exists and is complete, the $ A _ {i,ac} $( the parts of the operators $ A _ {i} $ on $ H _ {i,ac} $) are unitarily equivalent. If $ A _ {1} $ and $ A _ {2} $ are self-adjoint operators on $ H $ and $ A _ {2} = A _ {1} + c \langle \cdot , f \rangle f $, where $ f \in H $ and $ c $ is real, then $ W _ \pm ( A _ {2} , A _ {1} ) $ and $ W _ \pm ( A _ {1} , A _ {2} ) $ 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 an orthogonal projector in the West.

An operator $ W : H \rightarrow H _ {1} $ is partially isometric if there is a closed linear subspace $ M $ of $ H $ such that $ \| W u \| = \| u \| $ for $ u \in M $ and $ W v = 0 $ for $ v \in M ^ \perp $, the orthogonal complement of $ M $; the set $ M $ is called the initial set of $ W $ and $ M _ {1} = W ( M) $ the final set of $ W $.