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''with respect to a given orthogonal decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016680/b0166801.png" /> of a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016680/b0166802.png" />''
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A linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016680/b0166803.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016680/b0166804.png" /> which leaves each of the subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016680/b0166805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016680/b0166806.png" />, invariant. The spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016680/b0166807.png" /> is the closure of the union of the spectra of the  "blocks"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016680/b0166808.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016680/b0166809.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016680/b01668010.png" />. A block-diagonal operator in the broad sense of the word is an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016680/b01668011.png" /> of multiplication by a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016680/b01668012.png" /> in the direct integral of Hilbert spaces
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<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/b/b016/b016680/b01668013.png" /></td> </tr></table>
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''with respect to a given orthogonal decomposition  $  H = \sum _ {k \geq  1 }  \oplus H _ {k} $
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of a Hilbert space  $  H $''
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016680/b01668014.png" /> is a linear operator acting on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016680/b01668015.png" />. Each operator which commutes with a normal operator is a block-diagonal operator with respect to the spectral decomposition of this operator. See also [[Diagonal operator|Diagonal operator]].
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A linear operator  $  A $
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on  $  H $
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which leaves each of the subspaces  $  H _ {k} $,
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$  k \geq  1 $,
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invariant. The spectrum of  $  A $
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is the closure of the union of the spectra of the  "blocks"   $  A \mid  _ {H _ {k}  } = A _ {k} $,
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$  k \geq  1 $,
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$  \| A \| = \sup  _ {k \geq  1 }  \| A _ {k} \|  $.  
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A block-diagonal operator in the broad sense of the word is an operator  $  A $
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of multiplication by a function  $  \lambda $
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in the direct integral of Hilbert spaces
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$$
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H  =  \int\limits _ { M } \oplus H (t)  d \mu (t) ,\ \
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( A f ) (t)  =  \lambda (t) f (t) ,\  t \in M .
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$$
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Here  $  \lambda (t) $
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is a linear operator acting on the space $  H (t) $.  
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Each operator which commutes with a normal operator is a block-diagonal operator with respect to the spectral decomposition of this operator. See also [[Diagonal operator|Diagonal operator]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Naimark,  "Normed rings" , Reidel  (1984)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Naimark,  "Normed rings" , Reidel  (1984)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.R. Halmos,  "A Hilbert space problem book" , Springer  (1982)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.R. Halmos,  "A Hilbert space problem book" , Springer  (1982)</TD></TR></table>

Latest revision as of 10:59, 29 May 2020


with respect to a given orthogonal decomposition $ H = \sum _ {k \geq 1 } \oplus H _ {k} $ of a Hilbert space $ H $

A linear operator $ A $ on $ H $ which leaves each of the subspaces $ H _ {k} $, $ k \geq 1 $, invariant. The spectrum of $ A $ is the closure of the union of the spectra of the "blocks" $ A \mid _ {H _ {k} } = A _ {k} $, $ k \geq 1 $, $ \| A \| = \sup _ {k \geq 1 } \| A _ {k} \| $. A block-diagonal operator in the broad sense of the word is an operator $ A $ of multiplication by a function $ \lambda $ in the direct integral of Hilbert spaces

$$ H = \int\limits _ { M } \oplus H (t) d \mu (t) ,\ \ ( A f ) (t) = \lambda (t) f (t) ,\ t \in M . $$

Here $ \lambda (t) $ is a linear operator acting on the space $ H (t) $. Each operator which commutes with a normal operator is a block-diagonal operator with respect to the spectral decomposition of this operator. See also Diagonal operator.

References

[1] M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian)

Comments

References

[a1] P.R. Halmos, "A Hilbert space problem book" , Springer (1982)
How to Cite This Entry:
Block-diagonal operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Block-diagonal_operator&oldid=46086
This article was adapted from an original article by N.K. Nikol'skiiB.S. Pavlov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article