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An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d0315001.png" /> defined on the (closed) linear span of a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d0315002.png" /> in a normed (or only locally convex) space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d0315003.png" /> by the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d0315004.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d0315005.png" /> and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d0315006.png" /> are complex numbers. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d0315007.png" /> is a continuous operator, one has
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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/d/d031/d031500/d0315008.png" /></td> </tr></table>
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d0315009.png" /> is a Banach space, this condition is equivalent to the continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d03150010.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d03150011.png" /> is an unconditional basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d03150012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d03150013.png" /> is an orthonormal basis in a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d03150014.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d03150015.png" /> is a normal operator, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d03150016.png" />, while the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d03150017.png" /> coincides with the closure of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d03150018.png" />. A normal and completely-continuous operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d03150019.png" /> is a diagonal operator in the basis of its own eigen vectors; the restriction of a diagonal operator (even if it is normal) to its invariant subspace need not be a diagonal operator; given an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d03150020.png" />, any normal operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d03150021.png" /> on a separable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d03150022.png" /> can be represented as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d03150023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d03150024.png" /> is a diagonal operator, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d03150025.png" /> is a completely-continuous operator and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d03150026.png" />.
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An operator $  D $
 +
defined on the (closed) [[Linear closure|linear span]] of a basis  $  \{ e _ {k} \} _ {k \geq  1 }  $
 +
in a normed (or only locally convex) space $  X $
 +
by the equations  $  De _ {k} = \lambda _ {k} e _ {k} $,  
 +
where  $  k \geq  1 $
 +
and where $  \lambda _ {k} $
 +
are complex numbers. If  $  D $
 +
is a continuous operator, one has
  
A diagonal operator in the broad sense of the word is an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d03150027.png" /> of multiplication by a complex function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031500/d03150028.png" /> in the direct integral of Hilbert spaces
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$$
 +
\sup _ {k \geq  1 }  | \lambda _ {k} |  < + \infty .
 +
$$
  
<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/d/d031/d031500/d03150029.png" /></td> </tr></table>
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If  $  X $
 +
is a Banach space, this condition is equivalent to the continuity of  $  D $
 +
if and only if  $  \{ e _ {k} \} _ {k \geq  1 }  $
 +
is an unconditional basis in  $  X $.
 +
If  $  \{ e _ {k} \} _ {k \geq  1 }  $
 +
is an orthonormal basis in a Hilbert space  $  H $,
 +
then  $  D $
 +
is a normal operator, and  $  \| D \| = \sup _ {k \geq  1 }  | \lambda _ {k} | $,
 +
while the spectrum of  $  D $
 +
coincides with the closure of the set  $  \{ {\lambda _ {k} } : {k = 1 , 2 , .  .  . } \} $.
 +
A normal and completely-continuous operator  $  N $
 +
is a diagonal operator in the basis of its own eigen vectors; the restriction of a diagonal operator (even if it is normal) to its invariant subspace need not be a diagonal operator; given an  $  \epsilon > 0 $,
 +
any normal operator  $  N $
 +
on a separable space  $  H $
 +
can be represented as  $  N = D + C $,
 +
where  $  D $
 +
is a diagonal operator,  $  C $
 +
is a completely-continuous operator and  $  \| C \| < \epsilon $.
 +
 
 +
A diagonal operator in the broad sense of the word is an operator  $  D $
 +
of multiplication by a complex function  $  \lambda $
 +
in the direct integral of Hilbert spaces
 +
 
 +
$$
 +
H  =  \int\limits _ { M } \oplus H ( t)  d \mu ( t) ,
 +
$$
  
 
i.e.
 
i.e.
  
<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/d/d031/d031500/d03150030.png" /></td> </tr></table>
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$$
 +
( D f  )( t)  = \lambda ( t) f ( t) ,\  t \in M ,\  f \in H .
 +
$$
  
 
Cf. [[Block-diagonal operator|Block-diagonal operator]].
 
Cf. [[Block-diagonal operator|Block-diagonal operator]].
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Singer,  "Bases in Banach spaces" , '''1''' , Springer  (1970)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Wermer,  "On invariant subspaces of normal operators"  ''Proc. Amer. Math. Soc.'' , '''3''' :  2  (1952)  pp. 270–277</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.D. Berg,  "An extension of the Weyl–von Neumann theorem to normal operators"  ''Trans. Amer. Math. Soc.'' , '''160'''  (1971)  pp. 365–371</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Singer,  "Bases in Banach spaces" , '''1''' , Springer  (1970)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Wermer,  "On invariant subspaces of normal operators"  ''Proc. Amer. Math. Soc.'' , '''3''' :  2  (1952)  pp. 270–277</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.D. Berg,  "An extension of the Weyl–von Neumann theorem to normal operators"  ''Trans. Amer. Math. Soc.'' , '''160'''  (1971)  pp. 365–371</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 19:55, 27 February 2021


An operator $ D $ defined on the (closed) linear span of a basis $ \{ e _ {k} \} _ {k \geq 1 } $ in a normed (or only locally convex) space $ X $ by the equations $ De _ {k} = \lambda _ {k} e _ {k} $, where $ k \geq 1 $ and where $ \lambda _ {k} $ are complex numbers. If $ D $ is a continuous operator, one has

$$ \sup _ {k \geq 1 } | \lambda _ {k} | < + \infty . $$

If $ X $ is a Banach space, this condition is equivalent to the continuity of $ D $ if and only if $ \{ e _ {k} \} _ {k \geq 1 } $ is an unconditional basis in $ X $. If $ \{ e _ {k} \} _ {k \geq 1 } $ is an orthonormal basis in a Hilbert space $ H $, then $ D $ is a normal operator, and $ \| D \| = \sup _ {k \geq 1 } | \lambda _ {k} | $, while the spectrum of $ D $ coincides with the closure of the set $ \{ {\lambda _ {k} } : {k = 1 , 2 , . . . } \} $. A normal and completely-continuous operator $ N $ is a diagonal operator in the basis of its own eigen vectors; the restriction of a diagonal operator (even if it is normal) to its invariant subspace need not be a diagonal operator; given an $ \epsilon > 0 $, any normal operator $ N $ on a separable space $ H $ can be represented as $ N = D + C $, where $ D $ is a diagonal operator, $ C $ is a completely-continuous operator and $ \| C \| < \epsilon $.

A diagonal operator in the broad sense of the word is an operator $ D $ of multiplication by a complex function $ \lambda $ in the direct integral of Hilbert spaces

$$ H = \int\limits _ { M } \oplus H ( t) d \mu ( t) , $$

i.e.

$$ ( D f )( t) = \lambda ( t) f ( t) ,\ t \in M ,\ f \in H . $$

Cf. Block-diagonal operator.

References

[1] I.M. Singer, "Bases in Banach spaces" , 1 , Springer (1970)
[2] J. Wermer, "On invariant subspaces of normal operators" Proc. Amer. Math. Soc. , 3 : 2 (1952) pp. 270–277
[3] I.D. Berg, "An extension of the Weyl–von Neumann theorem to normal operators" Trans. Amer. Math. Soc. , 160 (1971) pp. 365–371

Comments

For the notion of an unconditional basis see Basis.

For diagonal operators in the broad sense (and the corresponding notion of a diagonal algebra) see [a1].

References

[a1] M. Takesaki, "Theory of operator algebras" , 1 , Springer (1979) pp. 259, 273
[a2] P.R. Halmos, "A Hilbert space problem book" , Springer (1982)
How to Cite This Entry:
Diagonal operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diagonal_operator&oldid=17619
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