Difference between revisions of "Similar operators"
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| − | + | Operators $ S $ | |
| + | and $ T $( | ||
| + | not necessarily bounded) on a Banach space $ X $ | ||
| + | for which there exists a bounded operator $ U $ | ||
| + | on $ X $ | ||
| + | having a bounded inverse and such that the following relation applies: | ||
| − | This concept is an example of the concept of similar mappings. Let | + | $$ |
| + | S = U ^ {-} 1 TU. | ||
| + | $$ | ||
| + | |||
| + | If $ U $ | ||
| + | is a [[Unitary operator|unitary operator]], then $ S $ | ||
| + | and $ T $ | ||
| + | are said to be unitarily equivalent. | ||
| + | |||
| + | This concept is an example of the concept of similar mappings. Let $ f $ | ||
| + | and $ g $ | ||
| + | be two mappings of a set $ X $ | ||
| + | into itself. If there is a [[Bijection|bijection]] $ U: X \rightarrow X $ | ||
| + | such that $ Uf = gU $, | ||
| + | then these mappings are said to be similar. Attempts have been made to give a definition of similarity for mappings from one set $ X $ | ||
| + | into another $ Y $; | ||
| + | for example, such mappings are called similar if there exist bijections $ U $ | ||
| + | and $ V $ | ||
| + | of the sets $ X $ | ||
| + | and $ Y $ | ||
| + | into themselves such that $ Vf = gU $. | ||
Latest revision as of 08:13, 6 June 2020
Operators $ S $
and $ T $(
not necessarily bounded) on a Banach space $ X $
for which there exists a bounded operator $ U $
on $ X $
having a bounded inverse and such that the following relation applies:
$$ S = U ^ {-} 1 TU. $$
If $ U $ is a unitary operator, then $ S $ and $ T $ are said to be unitarily equivalent.
This concept is an example of the concept of similar mappings. Let $ f $ and $ g $ be two mappings of a set $ X $ into itself. If there is a bijection $ U: X \rightarrow X $ such that $ Uf = gU $, then these mappings are said to be similar. Attempts have been made to give a definition of similarity for mappings from one set $ X $ into another $ Y $; for example, such mappings are called similar if there exist bijections $ U $ and $ V $ of the sets $ X $ and $ Y $ into themselves such that $ Vf = gU $.
How to Cite This Entry:
Similar operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Similar_operators&oldid=48699
Similar operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Similar_operators&oldid=48699
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article