View source for Block-diagonal operator

''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" />''

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

<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>

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]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Naimark,   "Normed rings" , Reidel  (1984)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.R. Halmos,   "A Hilbert space problem book" , Springer  (1982)</TD></TR></table>