View source for Block-diagonal operator

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''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|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>