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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c0238201.png" /></td> </tr></table>
+
$$
 +
W _ {+} ( A _ {2} , A _ {1} )  = s - \lim\limits _ {t \rightarrow x }  e ^ {it A _ {2} - it A _ {1} } P _ {1} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c0238202.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c0238203.png" /> are self-adjoint operators on a separable Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c0238204.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c0238205.png" /> is an ortho-projector into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c0238206.png" />, and such that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c0238207.png" /></td> </tr></table>
+
$$
 +
\{ {W _ {+} ( A _ {2} , A _ {1} ) x } : {
 +
\| W _ {+} ( A _ {2} , A _ {1} ) x \| =
 +
\| x \| } \}
 +
= \
 +
H _ {2,ac} .
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c0238208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c0238209.png" />, is the set of all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382010.png" /> that are spectrally absolutely continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382011.png" />, i.e. for which the spectral measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382012.png" /> of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382013.png" /> is absolutely continuous with respect to the Lebesgue measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382014.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382015.png" />, or the analogously defined operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382016.png" />, exists and is complete, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382017.png" /> (the parts of the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382018.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382019.png" />) are unitarily equivalent. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382021.png" /> are self-adjoint operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382025.png" /> is real, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382027.png" /> exist and are complete.
+
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 and orthogonal projector in the West.
 
An ortho-projector is usually called and orthogonal projector in the West.
  
An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382028.png" /> is partially isometric if there is a closed linear subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382029.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382030.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382031.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382033.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382034.png" />, the orthogonal complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382035.png" />; the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382036.png" /> is called the initial set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382038.png" /> the final set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023820/c02382039.png" />.
+
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 $.

Revision as of 17:45, 4 June 2020


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 and 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=46420
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article