Difference between revisions of "Complete operator"
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A generalized wave operator, i.e. a partially isometric operator defined by | 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 | + | 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 | + | 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 | + | 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==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> T. Kato, "Perturbation theory for linear operators" , Springer (1966) pp. Chapt. X Sect. 3</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> T. Kato, "Perturbation theory for linear operators" , Springer (1966) pp. Chapt. X Sect. 3</TD></TR></table> | ||
| − | |||
| − | |||
====Comments==== | ====Comments==== | ||
| − | An ortho-projector is usually called | + | An ortho-projector is usually called an orthogonal projector in the West. |
| − | An operator | + | 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 $. | ||
Latest revision as of 18:19, 10 October 2023
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 $.
How to Cite This Entry:
Complete operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_operator&oldid=12208
Complete operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_operator&oldid=12208
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article