Difference between revisions of "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==== | ====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> | ||
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| − | |||
====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=13450
Block-diagonal operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Block-diagonal_operator&oldid=13450
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