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

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 $W$.