View source for Diagonal operator

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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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" />.

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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i.e.

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Cf. [[Block-diagonal operator|Block-diagonal operator]].

====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>



====Comments====
For the notion of an unconditional basis see [[Basis|Basis]].

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Takesaki,   "Theory of operator algebras" , '''1''' , Springer  (1979)  pp. 259, 273</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Halmos,   "A Hilbert space problem book" , Springer  (1982)</TD></TR></table>